An Exposition on The Laws of Finite Pointed Groups
|
|
- Verity Marion Cannon
- 5 years ago
- Views:
Transcription
1 An Exposton on The Laws of Fnte Ponted Groups Justn Laverdure August 21, 2017 We outlne the detals of The Laws of Fnte Ponted Groups [1], wheren a fnte ponted group s constructed wth no fnte bass for ts laws. 1 Introducton 1.1 Defntons Defne a word on the countably nfnte alphabet {x = x 0, x 1, x 2,... } to be a strng of sad symbols, along wth ther formal nverses. That s, such a word s an element of the free group generated by {x 0, x 1, x 2,..., }. Gven a group G, say that a word w s a unversal law (or more often smply a law ) of G f, gven any assgnment of group elements g 0, g 1, g 2,... to the symbols x 0, x 1, x 2,..., the correspondng element of G s trval. For example, G s an abelan group f and only f x 1 x 2 x 1 1 x 1 2 s a law of G. We say that a ponted group (G, g) s a group G along wth a dstngushed element 1 g G. A homomorphsm of ponted groups φ : (G, g) (H, h) s a homomorphsm of groups φ : G H such that φ(g) = h. That s, homomorphsms must take dstngushed elements to dstngushed elements. A sub-ponted-group (H, h) of a ponted group (G, g) s a subgroup H G, whch addtonally s a ponted group such that h = g. That s, sub-ponted-groups are just subgroups contanng the dstngushed element. Thus, the sub-ponted-group generated by a subset S G of cardnalty S = ν s the subgroup generated by S {g}; ths sub-ponted-group s sad to be ν-generated (as a ponted group). It s a fact that the somorphsm theorems for groups hold equally well for ponted groups [4]. These ponted groups too have a noton of a law; the only dfferences wth the pror defnton 1 Sometmes, a ponted group wll be such wthout havng ts dstngushed element be explctly mentoned, f t can be nferred from context (especally f g has been mentoned already). Ths s about as dangerous as mentonng a rng wthout explctly mentonng ts addtve dentty. 1
2 are that we add a letter y to the alphabet from whch our words are constructed, and that we requre y to be nterpreted as g. For example, (G, g) has g Z(G) f and only f xyx 1 y 1 s a law of (G, g). Note how, wth ths addton to our language, we can talk about certan propertes held by g alone. Certan laws mply others; for example, f squares commute (.e. the law x 2 1 x2 2 x 2 1 x 2 2 ), then fourth powers commute too. Thus, we mght ask: gven a group G, what are the sets of laws of G from whch all laws of G may be deduced? Such a set of laws s a bass for the laws of G. Certanly the set of all laws s a bass for tself; we mght ask nstead, when can we fnd a smaller, more tractable bass? When can we fnd a fnte bass? A class of groups G s a varety f t s defned by some set of laws; that s, f there exsts some set W of words such that G G f and only f, for every w W, w s a law of G. It turns out that we may talk about the varety generated by a class of groups H: the smallest varety contanng every H H. Call ths varety Var(H). Fact: Var(H) s closed under takng homomorphc mages, subgroups, and (arbtrary) cartesan powers. So, f a group H s obtaned from a group H H by takng mages, subgroups, and powers (not necessarly n that order), then H Var(H). Analogous facts hold for varetes of ponted groups. General facts about the theory of varetes may be found n [4]. 1.2 Outlne When G s fnte, a theorem of Oates and Powell [3] tells us that a fnte bass always exsts. We now ask an analogous queston for ponted groups: does every fnte ponted group have a fnte bass for ts laws? Ths tme the answer s no, a result of Roger Bryant [1]. We follow the counter-example of Roger Bryant n constructng a fnte ponted group (P, p) wth no fnte bass for ts laws. Ths ponted group (P, p) has a one-parameter famly of words w n = w n (x 1,..., x n, y) as laws 2 ; we frst construct a one-parameter famly of ponted groups {(Q n, q n )} n N, such that each (Q n, q n ) does not have w 2 n as a law. Therefore, n partcular, no (Q n, q n ) s n the varety generated by (P, p). Each (Q n, q n ) wll turn out to addtonally have the property that every (n 1)-generator sub-ponted-group s n the varety generated by (P, p). Therefore, (P, p) wll not have have a fnte bass for ts laws; f t dd, then the number of 2 Among others, of course. The w n are not an exhaustve lst. 2
3 varables appearng across ths fnte bass would be fnte (say, there would be m of these varables), and hence Q m+1 (beng outsde of Var(P, p)) would have some a 1,..., a m Q m+1 whch fal some word w B. Yet, the m-generator sub-ponted-group a 1,..., a m, q k Q m+1 lands n the varety generated by (P, p), and hence would satsfy w, a contradcton. For a general reference on fnte groups, see [2]. 2 Constructng the (Q n, q n ) 2.1 Constructng the group A Let s see the (Q n, q n ). Fx a postve nteger n, and defne A = A n to be a (multplcatve) elementary abelan 2-group generated by bass elements e 1,..., e n. That s, A := e 1,..., e n e 2 = 1, [e, e j ] = 1. So, A has order 2 n, and each element a of A s wrtten unquely as n a = e ɛ, =1 wth each ɛ ether 0 or 1. It wll later be useful to now observe that each a, actng on A by leftor rght-multplcaton, s a permutaton of order 2. Beng elementary abelan, A s automatcally a vector space over the feld wth two elements F 2 = Z/2Z. Thus, we may speak of lnearly ndependent elements of A, of a bass for A, and so on, whch we shall do over the course of ths paper. It may also be helpful to realze that A has a rather large automorphsm group: GL n (F 2 ) Lemma 1. Aut(A) acts doubly transtvely on the non-trval elements of A. Proof. Say that we have non-trval elements a 1 a 2, a 1 a 2, and that we wsh to send a 1 to a 1 and a 2 to a 2. Because a 1 and a 2 are dstnct, non-trval elements of A, they are lnearly ndependent (because the only non-dentty multple of any a A s a 2 = 1 A). Thus, we may extend {a 1, a 2 } to a bass for A. Smlarly, {a 1, a 2 } s an ndependent set, and so may be extended to a bass. Thus, the desred automorphsm s gven by mappng the former bass to the latter. Morally, ths means that the e wll have no specal role to play, except n a countng argument 3
4 later. Secondly, ths means that we may, n an argument, wthout loss of generalty, pck any non-trval a A, and once we have done ths, agan pck, wthout loss of generalty, a non-trval, dstnct a A. We ll use ths repeatedly. 2.2 Constructng the group B Now, let β be a multplcatve cyclc group of order 3, and let B be the (multplcatve) elementary abelan 3-group generated by bass elements β a. At the moment, we regard these β a as formal symbols. That s, B := {β a } a A (β a ) 3 = 1, [β a, β a ] = 1. So, we may also vew B as β A, the set of β -valued functons on A, wth pontwse multplcaton. Ether way, ths s a group of order 3 2n, and each element b of B s wrtten unquely as b = a A(β a ) ka, where each k a Z/3Z = F 3. When we nterpret elements of B as functons, we have b(a) = k a. Let us vew β as beng an element of B by dentfyng β = β 1, where 1 A. Lke A, B s elementary abelan, and so s automatcally a vector space (n ths case over F 3 ). Therefore we shall freely use lnear algebra arguments also when dscussng B. 2.3 Constructng the acton of A on B by Now, let A act (on the rght) on the set of bass elements {β a } a A, permutng the bass elements β a a 0 := β aa 0. Observe that ths notaton makes sense, n the sense that we may vew β a as ether the prmtve symbol β a, whch generated B n the frst place, or as β acted on by a. Now, let us extend ths multplcatvely (and therefore lnearly) to the whole of B, defnng b a 0 = b a 0 by ( ) (β a ) ka a 0 := a A(β aa 0 ) ka. a A Thus, the acton of A on B s fathfully gven by a set of permutatons of the bass elements β a. Thus, unlke n A, the bass elements of B are dstngushed from other elements of B, specfcally by the acton of A on B. 4
5 2.4 A tangental observaton about B We may also see that (B, ) s somorphc, as a group, to the group rng (F 3 A, +), va a A(β a ) ka k a a, a A and that we may regard each element of B unquely as β a A kaa := a A(β a ) ka, off-loadng all of the nformaton about an element b = β r B nto the exponent r, and so that the above somorphsm becomes β r r. Accordngly, the rght-acton of A on B extends to an acton of the group algebra F 3 A on B, gven by β a0 k a a = (β a0a ) ka, a A a A and extendng multplcatvely. Thus, A acts by automorphsms of B, and generally, elements of F 3 A act by endomorphsms. Under the somorphsm (B, ) (F 3 A, +), ths looks lke k a a a 0 = k a aa 0. a A a A That s, the acton of A on B s just (rght-)multplcaton n F 3 A. 2.5 Constructng the group W Now, consder the semdrect product W := B A = BA (wth respect to the aforementoned acton of A on B), whch s also the wreath product β A. So, W s a group of order 2 n 3 2n. As wth any semdrect product, every element of ths group s wrtten unquely as ba, for some b B and a A. Further, as wth any semdrect product, we have the equaton ab = b a 1 a, whch n our case (where a 1 = a) s more smply stated as ab = b a a. So, we may further nterpret b a as b a = aba 1 = a 1 ba B W. 5
6 Ths s a specfc case of the prncple: whenever a group G acts (on the rght) on another group H, we shall wrte h g for the mage of h under the acton of g. Ths wll always be conjugaton wthn a semdrect product HG = H G. 2.6 Countng maxmal subgroups Now, A, as an elementary abelan 2-group, s automatcally a F 2 -vector space; maxmal subgroups are then just subspaces of dmenson n 1. Further, by defnng an nner product on A by e, e j := δ j and extendng blnearly, where δ j s the Kronecker delta, we see that, for all k, subspaces S A of dmenson k are n bjecton wth subspaces of dmenson n k by S S, where S := {a A : a, S = 0}. So, f we would lke to count the number of maxmal subgroups of A, t s suffcent to count the number of 1-dmensonal subspaces {1, a} A. But, these are n bjecton wth the non-trval elements a A, of whch there are 2 n 1. So, suppose that the maxmal subgroups M of A are gven by M 0, M 1,..., M 2r, where r := 2 n 1 1. One mght note that the automorphsm group of A also acts doubly transtvely on these maxmal subgroups; that s, the M are also hghly nterchangeable. Specfcally, we have the same stuaton as that for non-trval elements a A, as establshed n Lemma 1: n an argument, we may wthout loss of generalty pck any M, and then wthout loss of generalty pck any dstnct M. 2.7 Certan subgroups of commutators of W Gven a maxmal subgroup M of A, consder now the commutator subgroup [B, M ] = b 1 m 1 bm : b B, m M. Now, we have b 1 m 1 bm = b 1 b m, and b = a A (βa ) ka. The β a commute, so that [B, M ] (β a ) 1 β am : a A, m M, 6
7 because each b s a product of the commutng β a s. commutator [β a, m] [B, M ], so we have Conversely, each (β a ) 1 β am s n fact the [B, M ] = (β a ) 1 β am : a A, m M. Now, for each m M whch s not 1 (of whch there are 2 n 1 1), consder the element β m β 1. Smlarly, fx some a 0 A \ M, and consder the 2 n 1 1 elements β ma 0 (β a 0 ) 1, where m ranges agan over the non-trval elements of M. We ll see n a moment that ths set S := {β m β 1, β ma 0 (β a 0 ) 1 : m M \ {1}} of sze 2 n 2 s a bass for [B, M ], so that each [B, M ] s a group of order 3 2n 2 and ndex 3 2 n B. () That S s ndependent may be seen by consderng the matrx , where each block has 2 n 1 rows and 2 n 1 1 columns. It s easy to see that ths matrx has full rank: 2 n 2. Now, we may nterpret the columns of ths matrx as the vectors n S, represented wth respect to the ordered bass ((β m ) m M \{1}, (β a0m ) m M \{1}), and n partcular the frst row corresponds to β, and row number 2 n corresponds to β a 0. Therefore, [B, M ] has dmenson at least 2 n 2. () Secondly, S spans because [B, M ] s n the kernel of the lnear mappng Ψ : B F 3 F 3, 7
8 where Ψ(b) = (Ψ 1 (b), Ψ 2 (b)) and ( ) Ψ 1 (β a ) ka := k a a A a M ( ) Ψ 2 (β a ) ka := k a. a A a A\M Ths follows from the fact that the generators (β a ) 1 β am of [B, M ] are n the kernel of Ψ. It s easy to see that ths mappng has mage of dmenson two, and therefore that ker f has dmenson 2 n 2, so that [B, M ] ker f has dmenson no more than 2 n 2. Note the general form of ths argument. We break up A nto a partton X = {aa : a A} gven by the cosets of some subgroup A A, exhbt lnearly ndependent vectors n B that, f we nterpret them as functons b : A β, have support contaned exclusvely n one element of ths partton, and fnd lnear functonals f whch only depend on components of the vector correspondng to some element of the partton, that s, functonals f of the form f(b) = f (b aa ), where aa s some coset of A and f s some affne functonal defned on aa. Arguments of ths form wll be used n Secton 4 and Certan quotents W of W correspondng (loosely) to sad subgroups Now, each [B, M ]M s a subgroup of W, as, gven an arbtrary generator b 1 b m [B, M ] and an arbtrary n M, (b 1 b m ) n = (b n ) 1 (b n ) m [B, M ]M, because b n B. Thus, [B, M ]M = M [B, M ]; that s, M acts on [B, M ] by conjugaton, and [B, M ]M = [B, M ] M s n fact a semdrect product tself. Thus, each [B, M ]M s a subgroup of order 2 n 1 3 2n 2 and ndex n W. In fact, each s a normal subgroup, as, gven an arbtrary generator b 1 b m n [B, M ]M and an arbtrary b 0 a 0 BA, we have (b 1 b m n) b 0a 0 = [ (b a 0 ) 1 (b a 0 ) m][ (b a 0 ) 1 (b a 0 ) n] n [B, M ]M where we ve used the fact that B and A act on themselves trvally by conjugaton (.e. B and A are abelan), and that a b = [b, a]a for any a A and b B. Thus, we may consder the quotent W = W/[B, M ]M 8
9 along wth the natural homomorphsm φ : W W. Now, n the mage of φ, the subgroup M collapses down to 1, the bass elements {β m : m M } are all dentfed, as are the elements {β a0m : m M }. Thus, W s somorphc to W n the n = 1 case,.e. W (Z/3Z Z/3Z) Z/2Z, where Z/2Z acts by nterchangng (1, 0) and (0, 1). Say that β := φ (β) and β := φ (β a 0 ) correspond to these elements, and further let us defne ξ = φ (a 0 ). Thus, W = β, β, ξ. 2.9 Auxlary groups Z Z and actons on them by the W For each, consder two copes of the quaternon group (of order 2 3 ), Z = ζ, η, θ, ι, ζ 2 = 1, η 2 = θ 2 = ι 2 = η θ ι = ζ and Z generated smlarly by elements ζ, η, θ, ι, wth smlar relatons. These ζs, ηs, θs, and ιs are, of course, analogues of 1,, j, k Q 8 C. Let each W act on the correspondng Z Z (on the rght), where: a) β acts by permutng η, θ, ι Z and fxes Z pont-wse, b) conversely, β permutes η, θ, ι Z and fxes Z pont-wse, and c) ξ permutes Z and Z,.e. (a, b) (b, a). Observe that the actons of β and β are commutng, exponent-3 bjectons, whch are conjugate by the (nvolutve) acton of ξ. These are exactly the relatons of W ; thus, ths acton s well-defned (and fathful) Constructng the groups C and the actons on them by the W Now, consder the central, non-trval elements ζ Z and ζ Z, along wth ζ ζ Z Z. Ths latter element s agan central, and so we can consder the quotent C := Z Z ζ ζ, a group of order 2 5. We wll abuse notaton, and refer to elements of C as tuples. Let γ C (somewhat arbtrarly) be the mage of η Z Z, so that γ has order 4. Also, let δ := γ 2, so that δ s the mage of both ζ and ζ n C. So, computng n C, we have that δ = (ζ, 1) = (1, ζ ) C 9
10 s a central element of order 2, and s addtonally a commutator, as j = j Q 8, so, agan computng n C, δ = [(η, 1), (θ, 1)] C. In fact, δ alone s the centre and commutator subgroup of C, whch may be seen explctly: 1 Q 8 s the centre and commutator subgroup of Q 8, so that (1, 1), ( 1, 1) s the centre and commutator subgroup of Q 8 Q 8, and the quotent of ths group by ( 1, 1) s exactly analogous to our case. Now, because the aforementoned acton of W on Z Z fxes ζ ζ, ths acton yelds up a well-defned quotent acton on C Facts and computatons We wll later need to observe that γ β = (1 β, η β ) = (1, η ) = γ C, that [γ, γ β ] = [(1, η ), (1 β β, η )] = [(1, η ), (1, θ )] = (1, ζ ) = δ C, and that, smlarly, whch all together mply that [γ, γ β 1 ] = [(1, η ), (1 β β, η 1 )] = [(1, η ), (1, ι )] = (1, ζ ) = δ C, [γ, γ β β ] = [γ, γ (β β ) 1 ] = δ C. Further, we should know that δ ba = (δ φ (b) ) φ(a) = δ because both β and β fx δ, as does ξ. Thus, each δ s fxed by the acton of W. At ths pont, we may mostly strke from our mnds the ntty-grtty of the quaternon groups Constructng the groups C and D and the actons on C and C/D by W Let 2r C := =0 C 10
11 be the drect product of the groups C (and therefore a group of order 32 2r+1 = 2 5 2n 5 ), and let us consder the subgroup D C generated by the elements δ δ j, for 0, j 2r. That s, we defne D := δ δ j : 0, j 2r. Note that the group D := δ : 0 2r C s generated by (2 n 1)-many commutng nvolutons, and s therefore elementary abelan of order 2 2n 1. Vewng the δ as an F 2 -bass of D, we have the lnear functonal f : D F 2 gven by δ 1 for every 0 2r and extendng lnearly; the mage of an element d D we may vew as ts party. In partcular, we also have the restrcton f D of f to D, and we may talk about the party of these elements. In fact, lookng agan at the generatng set of D, every element of D has even party; better yet, D s exactly the elements of D wth even party. Thus, D = f 1 ({0}), and so D has order 2 2n 2 and ndex 2 2n+2 3 n C. We wll later use the fact that δ 0 δ 2r D, as there are 2r + 1 such ndces (an odd number). Now, let W act on each C by pullng back each φ, that s, let us defne, for c C, w c := φ (w) c. Now, let us extend ths component-wse to an acton on all of C, that s, for c = (c ) 2r =0 C, w (c ) 2r =0 := (φ (w) c ) 2r = Constructng the ponted group (Q n, q n ) So, we may consder the semdrect product CW = C W, a group of order 2 5 2n +n 5 3 2n. Now, as we ve seen, each δ s fxed by W, so that all of D s, so that W nherts a quotent acton on C/D. Now we get to the frst punchlne: let Q n := (C/D)W, and gven the element γ := γ 0 γ 2r = (γ ) 2r =0 C, defne q n := (γ 1 D)β (C/D)W. Thus, we have our one-parameter famly of ponted groups (Q n, q n ), each respectvely of order 2 2n+2 +n 3 3 2n. 11
12 We now have three thngs remanng to show: that each word w 2 n s not a law of the correspondng (Q n, q n ), to construct the ponted group (P, p), and that every (n 1)-generator sub-ponted-group of (Q n, q n ) s n the varety generated by (P, p). Let s see these n ths order. 3 Falure of the law w 2 n n (Q n, q n ) We defne the words v n (x 1,..., x n, y) := y x1 y xn w n (x 1,..., x n, y) := [y 3, (y 3 ) vn ]. Now, let the elements of A be enumerated as α 1,..., α 2 n, and let ρ := (γ 1 β) α1 (γ 1 β) α 2 n = v 2 n(α 1,..., α 2 n, γ 1 β) CW σ := [(γ 1 β) 3, ((γ 1 β) 3 ) ρ ] = w 2 n(α 1,..., α 2 n, γ 1 β) CW. These elements ρ and σ are thus mages of the words v n and w n n CW. In partcular, t follows that w 2 n(α 1,..., α 2 n, (γ 1 D)β) = σd (C/D)W too s a value of w 2 n, whch s non-trval f and only f σ D, so t wll suffce to show the latter. Frst, we observed that γ β = γ, whch just means that γ and β commute. Further, ths mples that γ β = γ, by the coordnate-wse defnton of the acton of W on C. Thus, γ and β commute. But ths means that (γ 1 β) 3 = γ 3 β 3 = γ 3 = γ. In fact, we may generalze ths fact a lttle bt: gven any α A, we wll see that γ α and β α commute. Remember that C s a drect product of the C ; let us fx any such, and consder the th component of (γ α ) βα, whch s (γ φ (α ) ) φ (β α), agan usng the defnton of the acton of W on C. If α M, then φ (α) = 1 and φ (β α ) = β, so that (γ φ (α ) ) φ (β α) = (γ 1 ) β = γ. In the other case, when α M, we have φ (α) = ξ and φ (β α ) = β, so that (γ φ (α ) ) φ (β α) = (γ ξ )β = γ ξ. because γ ξ has trval frst co-ordnate. But, n ether case, the acton of φ (β α ) s always trval 12
13 on γ φ (α ). Thus, (γ φ (α ) ) φ (β α) = γ φ (α ) whenever 0 2r. Thus, (γ α ) βα = γ α,.e. γ α and β α commute. So, ρ = (γ 1 β) α1 (γ 1 β) α 2 n = (γ 1 ) α1 (γ 1 ) α 2 n β α1 β α 2 n. It s also true that γ commutes wth γ α, for any α (whch may be seen by agan lookng at each co-ordnate). Thus, γ (γ 1 ) α 1 (γ 1 ) α 2 n Ths mmedately mples that [γ, γ ρ ] = [γ, γ τ ]. = γ, so that γ ρ = γ τ, where τ := β α1 β α 2 n. Now, breakng down nto components, we may wrte [γ, γ τ ] = [γ 0, γ φ 0(τ) 0 ] [γ 2r, γ φ 2r(τ) 2r ]. And then, we observe that 2 k 1 (mod 3) f and only f k s even, and otherwse 2 k 2 (mod 3). Ths, along wth the fact that exactly half,.e. 2 n 1, of the elements of A are elements of M, shows that φ (τ) = φ (β α1 β α 2 n ) = (β β ) 2n 1, whch s β β or (β β ) 1 f n s odd or even, respectvely. Ether way, a fact we ve prevously observed. So, fnally But, as we ve observed prevously, δ 0 δ 2r D. [γ, γ φ (τ) ] = δ, σ = [γ, γ ρ ] = [γ, γ τ ] = δ 0 δ 2r. 4 A few prelmnares We d lke to move on to constructng (P, p) now, but we need two thngs frst: to outlne the strategy (whch s to see that every (n 1)-generator sub-ponted-group R Q n s n the varety generated by (P, p)), and a lemma (whch s the fact that the subgroups [B, M ]M W have trval ntersecton). We wll see that 3 R s a subgroup of a group (C/D)BM, whch s n turn a homomorphc mage of a group (C/E)BM. Ths latter group wll be somorphc to a subgroup of a group r k=0 Ω k, 3 In ths dscusson, we gnore the dstngushed elements n our ponted groups for notatonal clarty. 13
14 whch wll tself be a quotent of a power of P. Hence, we wll see that (R, q n ) HSP((P, p)) va a chan of Hs, Ss, and P s. Of course, because HSP Var, ths wll show the desred fact. Lemma 2. The subgroups [B, M ]M W have trval ntersecton. Proof. Because the M have trval ntersecton, t s suffcent to see that the [B, M ] have trval ntersecton. Recall that each [B, M ] s assocated wth the two lnear functonals Ψ, Ψ : B F 3, gven by ( ) Ψ (β a ) ka := k a, a A a M ( ) Ψ (β a ) ka := k a. a A a A\M As we ve observed prevously, [B, M ] = ker Ψ ker Ψ. For each coset of a subgroup αs A, we have the lnear functonal (β a ) ka k a. a A a αs If S has ndex k, say that ths map s type k. Suppose that b = (β a ) ka [B, M ]. a A 0 2r We wll argue that, f b s n the kernel of every lnear functonal of type k + 1, then t s n the kernel of every lnear functonal of type k, for 0 k n 2. Ths, along wth the fact that k a = k a + k a = 0 a A a M a A\M for some (and every) (because b s n every [B, M ]) wll mply that k a = k a0 = 0, a {a 0 } for every a 0 A. But, ths s the fact that b = 1. Suppose so, that s, that b s n the kernel of every lnear functonal of type k + 1. Let αs be an arbtrary coset of a subgroup S of ndex k. A/S s elementary abelan of order 2 n k, whch s at least 4. Thus, A/S contans a subgroup of order 4 contanng αs, necessarly somorphc to the Klen four-group. Suppose that ths subgroup s gven by {αs = α 1 S, α 2 S, α 3, α 4 S}. 14
15 Defne Σ p F 3 to be a α k ps a, for 1 p 4. Then, because the sx quanttes k a = Σ p + Σ q a α ps α qs are all 0 by our nductve hypothess, for 1 p < q 4, (and because 2 s nvertble modulo 3), ths mples by a lttle lnear algebra that each Σ p = 0; n partcular, Σ 1 = k a = 0. But, ths s the concluson of the nducton. a αs Thus, k a0 = 0, so that b = 1, so that 0 2r [B, M ] = {1}. 4.1 Step 1 n the HSP chan Now, let s construct the ponted group (P, p), keepng n mnd also our goal: to see that every (n 1)-generator sub-ponted-group of (Q n, q n ) s n the varety generated by (P, p). The dea s ths: supposng that (R, q n ) s such a sub-ponted-group of (Q n, q n ), then note Q n = (C/D)BA, a semdrect product, and n Q n we know C/D to be a normal subgroup. Smlarly, B s a normal subgroup of BA, agan a semdrect product. Thus, we have the projectons (C/D)BA BA A, and we consder ther composton χ : Q n A. Because (R, q n ) (by assumpton) s an (n 1)- generator ponted group, (χ(r), χ(q n )) (A, χ(q n )) s also such. In partcular, though, χ(q n ) = χ(γ 1 β) = 1, so that χ(r) s really just an (n 1)-generator subgroup of A (.e. one whch s trvally ponted). Now, because A s not (n 1)-generated (whch we may see by a vector space dmenson argument, say), χ(r) s a proper subgroup of A, and s therefore contaned n some maxmal subgroup M A. In partcular, (R, q n ) s a sub-ponted-group of ((C/D)BM, q n ), a group of order 2 2n+2 +n 4 3 2n Thus, a choce of R gves us a partcular M, but as we ve observed prevously n Lemma 1, because all of the M are hghly nterchangeable, t doesn t matter whch M we choose, n the sense that the groups we ll construct are all somorphc. 5 Constructng the group (P, p) Now we ll construct (P, p) for real. 15
16 5.1 Parng the maxmal subgroups M of A Choose some maxmal subgroup M A. Now, gven any other maxmal subgroup S A, the ntersecton S M s always a subspace of co-dmenson two, so that A/(S M) s somorphc to the Klen four-group (.e. the unque F 2 -vector-space of dmenson 2). In the quotent, both M and S are sent to subgroups of order 2, and we can see that there s exactly one other such subgroup of the Klen four-group; call ts premage under the quotent map T. Thus, T S and T M = S M; set-theoretcally, we have T = S M (A \ (S M)). Thus, we have a parng S T of the maxmal subgroups of A not equal to M. So, let s suppose that the subgroups are enumerated as M 0,..., M 2r so that M = M 0, and M 2k 1 s pared wth M 2k, for 1 k r. We wll use the partcular fact that M 2k 1 M = M 2k M repeatedly. 5.2 Constructng the group E; step 2 n the HSP chan We defned earler the subgroup D of C generated by all products δ δ j ; let us now consder the subgroup E D generated only by consecutve products δ 2k 1 δ 2k, for 1 k r. That s, defne E := δ 2k 1 δ 2k : 1 k r, a group of order 2 r = 2 2n 1 1 and ndex 2 9 2n 1 4 n C. Thus, C/D s a quotent group of C/E, and n partcular ((C/D)BM, q n ) s a quotent pontedgroup of ((C/E)BM, (γ 1 E)β), the latter beng of order 2 9 2n 1 +n 5 3 2n. 5.3 Constructng the groups Ω k ; step 3 n the HSP chan Defne N 0 := [B, M 0 ]M 0 (whch we recall to be of order 2 n 1 3 2n 2 ), and for 1 k r, N k := [B, M 2k 1 ]M 2k 1 [B, M 2k ]M 2k. Then, because an ntersecton of normal subgroups s normal, all of the N k are normal subgroups of W. Further, because by smlar dmenson-countng arguments 4 to those n Secton 2.7, these 4 We can partton A nto the four cosets of M 2k 1 M 2k. Then, for each element X of ths partton, demonstrate 2 n 2 1 lnearly ndependent elements of B whose support s contaned n X, for a total of 2 n 4 such vectors. Further, there are four lnear functonals, of the form a A (βa ) ka a X ka the ntersecton of whose kernels contans [B, M 2k 1 ] [B, M 2k ]. 16
17 latter N k have order 2 n 2 3 2n 4. Proceedng smlarly, we defne the correspondng F 0 := C 0 (whch we recall to be order 2 5 ), and for 1 k r, F k = (C 2k 1 C 2k )/E k, where these E k are δ 2k 1 δ 2k, so that these F k are order 2 9. Because these δ 2k 1 δ 2k commute, we have E = E 1 E r, so that F 0 F 1 F r = C/E, or at least, these may be dentfed nnocuously. Yet another defnton: for 0 k r, let K k := (F 0 F 1 F k 1 F k+1 F r )N k (C/E)BM, so that K 0 s a group of order 2 9 2n 1 +n n 2 and ndex n (C/E)BM, and the other K k are order 2 9 2n 1 +n n 4 and ndex n (C/E)BM. The drect summand F 0 F 1 F k 1 F k+1 F r s a normal subgroup of the drect product C/E, as well as a normal subgroup of BM by the defnton of the acton of BA on C/E. Also, each N k s a normal subgroup of (C/E)BM by a smlar computaton to that n secton 2.8. So, each K k s a normal subgroup of (C/E)BM. Note how the F 0 F 1 F k 1 F k+1 F r have trval ntersecton; because the N k have trval ntersecton as well by Lemma 2, the K k themselves have trval ntersecton. For 0 k r, defne Ω k := ((C/E)BM)/K k as ponted groups. Let us take a moment to recall that, a pror, each Ω k depends on n as well; let us sgnfy ths by wrtng Ω k,n as needed. We have the map r : (C/E)BM k=0 Ω k (ce)bm (((ce)bm)k k ) r k=0 whose kernel s the ntersecton of the K k, whch we ve seen to be trval. somorphc to a sub-ponted-group of Ω 0 Ω 1 Ω r, va ths map. Thus, (C/E)BM s 17
18 5.4 Constructng the ponted group (P, p); the fnal step n the HSP chan Fnally, up to smorphsm, there s only one Ω 0,n, as well as only one Ω k,n across all k 1. Ths s because, n general, for a semdrect product G H and a normal subgroup G 1 H 1, we have where the acton on the rght s gven by G H G 1 H 1 G G 1 H H 1, (gg 1 ) hh 1 = g h G 1, f and only f [G, H 1 ] G 1. In our case, ths holds, as, gven any (q 1, q 2 ) C 2k 1, any a A, and any m, n M 2k 1, we have that (q 1, q 2 ) (βa ) 1 (β a ) mn = (q 1, q 2 ), because ether both a and am are n M 2k 1, or nether are; n ether case, (β a ) 1 (β a ) m and n both act trvally on C 2k 1. Thus, [(q 1, q 2 ), [β a, m]] = (q 1, q 2 ) 1 (q 1, q 2 ) (βa ) 1 (β a ) mn = (1, 1). The correspondng fact holds for C 2k. Thus, [C 2k 1 C 2k, [B, M 2k 1 ]M 2k 1 [B, M 2k ]M 2k ] = 1, and so [(C/E), N k ] F 0 F 1 F k 1 F k+1 F r. Thus, Ω 0 F 0 BM BM [B,M 0 ] and Ω k F k [B,M 2k 1 ] [B,M 2k ] for k 1. But, yet agan by elementary abelan-ness and dmenson-countng, these groups do not depend on n. So, because Ω k,n Ω k,n for non-zero k, k (and separately so for zero k), and because these Ω are ndependent of the choce of M, t s suffcent to take P := Ω 0 Ω 1, for an arbtrary, fxed n and M. So, P s a group of order Thus, r k=0 Ω k s a quotent of P r, a power of P. Thus, n summary, we have r R (C/D)BM (C/E)BM Ω k P r Var(P ), so that (R, q n ), the arbtrary (n 1)-generator sub-ponted-group of (Q n, q n ), les n the varety generated by (P, p). k=0 18
19 6 The laws w m of the ponted group (P, p) Now, we have just one task: show that, for every m N, the word w m s a law of (P, p). By the constructon of P, t s suffcent to show that w m s a law of each Ω k, and therefore to show that w m s a law of ((C/E)BM, (γ 1 E)β). Suppose we have arbtrary κ 1,..., κ m CBM, and consder now the elements g := v m (κ 1,..., κ m, γ 1 β) = (γ 1 β) κ1 (γ 1 β) κm, h := w m (κ 1,..., κ m, γ 1 β) = [(γ 1 β) 3, ((γ 1 β) 3 ) g ]. For reasons smlar to those mentoned n Secton 3, t s suffcent to observe that h E. that Because each κ j s an element of CBM = C (B M), we may let λ j CB and µ j M such κ j = λ j µ j for all 1 j m. Then, by a smlar calculaton as n Secton 3, we see that h = [γ, γ b ], where b = β µ1 β µm. In partcular, both γ and γ b are n C, so that h s too. Thus, because C = 2r =0 C s a drect product and by the defnton of the acton of b W on C, we may wrte and each [γ, γ φ (b) [γ, γ b ] = [γ 0, γ φ 0(b) 0 ] [γ 2r, γ φ 2r(b) 2r ], ] s n C. In fact, each [γ, γ φ (b) ] s ether δ or 1, by our observaton that the commutator subgroup of each C s δ. Now, φ (b) = φ (β µ1 β µm ) = β l β m l, where l s the number of ndces j for whch µ j M. For = 0, we have µ j M = M 0 for every 1 j m, so φ 0 (b) = β m 0, and so [γ 0, γ φ 0(b) 0 ] = [γ 0, γ βm 0 0 ] = [γ 0, γ 0 ] = 1. Otherwse, we recall the parng of M 2k 1 wth M 2k, for 1 k r. Because M 2k 1 M = M 2k M 19
20 and each µ j M (as we just observed), µ j M 2k 1 µ j M 2k, for any 1 k r. But ths means that, fxng some such k, for the same l. φ 2k 1 (b) = β l 2k 1 βm l 2k 1 φ 2k (b) = β l 2k βm l 2k Thus, [γ 2k 1, γ φ 2k 1(b) 2k 1 ] = δ 2k 1 f and only f [γ 2k, γ φ 2k(b) 2k ] = δ 2k. That s, h s a product of elements of the form δ 2k 1 δ 2k, for varous 1 k r, whch s exactly the fact that h E. 7 Appendx Below s a table detalng most of the groups whch appear n the constructon n the order of ther defnton, along wth ther orders. The bounds for ndces and k are omtted; ther values are the same as those n the man body of the paper. Group Order Group Order A 2 n (C/D)BM 2 2n+2 +n 4 3 2n M 2 n 1 E 2 2n 1 1 B 3 2n C/E 2 9 2n 1 4 W 2 n 3 2n (C/E)BM 2 9 2n 1 +n 5 3 2n [B, M ] 3 2n 2 N 0 2 n 1 3 2n 2 [B, M ]M 2 n 1 3 2n 2 N k 2 n 2 3 2n 4 W F C 2 5 F k 2 9 C 2 5 2n 5 K n 1 +n n 2 D 2 2n 1 K k 2 9 2n 1 +n n 4 D 2 2n 2 Ω C/D 2 2n+2 3 Ω k Q n 2 2n+2 +n 3 3 2n P
21 References [1] Roger M Bryant. The laws of fnte ponted groups. In: Bulletn of the London Mathematcal Socety 14.2 (1982), pp [2] Danel Gorensten. Fnte groups. Vol Amercan Mathematcal Soc., [3] Shela Oates and MB Powell. Identcal relatons n fnte groups. In: Journal of Algebra 1.1 (1964), pp [4] Hanamantagouda P Sankappanavar and Stanley Burrs. A course n unversal algebra. In: Graduate Texts Math 78 (1981). 21
SL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationFINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationPolynomials. 1 More properties of polynomials
Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a
More informationwhere a is any ideal of R. Lemma Let R be a ring. Then X = Spec R is a topological space. Moreover the open sets
11. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationProblem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?
Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2
More information2 More examples with details
Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and
More informationRestricted Lie Algebras. Jared Warner
Restrcted Le Algebras Jared Warner 1. Defntons and Examples Defnton 1.1. Let k be a feld of characterstc p. A restrcted Le algebra (g, ( ) [p] ) s a Le algebra g over k and a map ( ) [p] : g g called
More informationMath 594. Solutions 1
Math 594. Solutons 1 1. Let V and W be fnte-dmensonal vector spaces over a feld F. Let G = GL(V ) and H = GL(W ) be the assocated general lnear groups. Let X denote the vector space Hom F (V, W ) of lnear
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationALGEBRA HW 7 CLAY SHONKWILER
ALGEBRA HW 7 CLAY SHONKWILER 1 Whch of the followng rngs R are dscrete valuaton rngs? For those that are, fnd the fracton feld K = frac R, the resdue feld k = R/m (where m) s the maxmal deal), and a unformzer
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More informationMath 101 Fall 2013 Homework #7 Due Friday, November 15, 2013
Math 101 Fall 2013 Homework #7 Due Frday, November 15, 2013 1. Let R be a untal subrng of E. Show that E R R s somorphc to E. ANS: The map (s,r) sr s a R-balanced map of E R to E. Hence there s a group
More informationOn the partial orthogonality of faithful characters. Gregory M. Constantine 1,2
On the partal orthogonalty of fathful characters by Gregory M. Constantne 1,2 ABSTRACT For conjugacy classes C and D we obtan an expresson for χ(c) χ(d), where the sum extends only over the fathful rreducble
More informationFixed points of IA-endomorphisms of a free metabelian Lie algebra
Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp. 405 416. c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationALGEBRA MID-TERM. 1 Suppose I is a principal ideal of the integral domain R. Prove that the R-module I R I has no non-zero torsion elements.
ALGEBRA MID-TERM CLAY SHONKWILER 1 Suppose I s a prncpal deal of the ntegral doman R. Prove that the R-module I R I has no non-zero torson elements. Proof. Note, frst, that f I R I has no non-zero torson
More informationa b a In case b 0, a being divisible by b is the same as to say that
Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationINTERSECTION THEORY CLASS 13
INTERSECTION THEORY CLASS 13 RAVI VAKIL CONTENTS 1. Where we are: Segre classes of vector bundles, and Segre classes of cones 1 2. The normal cone, and the Segre class of a subvarety 3 3. Segre classes
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More information( 1) i [ d i ]. The claim is that this defines a chain complex. The signs have been inserted into the definition to make this work out.
Mon, Apr. 2 We wsh to specfy a homomorphsm @ n : C n ()! C n (). Snce C n () s a free abelan group, the homomorphsm @ n s completely specfed by ts value on each generator, namely each n-smplex. There are
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationREAL ANALYSIS I HOMEWORK 1
REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α
More informationRandom Walks on Digraphs
Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationCharacter Degrees of Extensions of PSL 2 (q) and SL 2 (q)
Character Degrees of Extensons of PSL (q) and SL (q) Donald L. Whte Department of Mathematcal Scences Kent State Unversty, Kent, Oho 444 E-mal: whte@math.kent.edu July 7, 01 Abstract Denote by S the projectve
More information28 Finitely Generated Abelian Groups
8 Fntely Generated Abelan Groups In ths last paragraph of Chapter, we determne the structure of fntely generated abelan groups A complete classfcaton of such groups s gven Complete classfcaton theorems
More informationHomework Notes Week 7
Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationOn intransitive graph-restrictive permutation groups
J Algebr Comb (2014) 40:179 185 DOI 101007/s10801-013-0482-5 On ntranstve graph-restrctve permutaton groups Pablo Spga Gabrel Verret Receved: 5 December 2012 / Accepted: 5 October 2013 / Publshed onlne:
More informationEdge Isoperimetric Inequalities
November 7, 2005 Ross M. Rchardson Edge Isopermetrc Inequaltes 1 Four Questons Recall that n the last lecture we looked at the problem of sopermetrc nequaltes n the hypercube, Q n. Our noton of boundary
More information20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.
20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed
More informationAppendix B. Criterion of Riemann-Stieltjes Integrability
Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for
More informationNOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules
NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules EVAN WILSON Quantum groups Consder the Le algebra sl(n), whch s the Le algebra over C of n n trace matrces together wth the commutator
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationLecture 7: Gluing prevarieties; products
Lecture 7: Glung prevaretes; products 1 The category of algebrac prevaretes Proposton 1. Let (f,ϕ) : (X,O X ) (Y,O Y ) be a morphsm of algebrac prevaretes. If U X and V Y are affne open subvaretes wth
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationSupplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso
Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016
U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and
More informationSpectral Graph Theory and its Applications September 16, Lecture 5
Spectral Graph Theory and ts Applcatons September 16, 2004 Lecturer: Danel A. Spelman Lecture 5 5.1 Introducton In ths lecture, we wll prove the followng theorem: Theorem 5.1.1. Let G be a planar graph
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationAnti-van der Waerden numbers of 3-term arithmetic progressions.
Ant-van der Waerden numbers of 3-term arthmetc progressons. Zhanar Berkkyzy, Alex Schulte, and Mchael Young Aprl 24, 2016 Abstract The ant-van der Waerden number, denoted by aw([n], k), s the smallest
More informationNotes on Frequency Estimation in Data Streams
Notes on Frequency Estmaton n Data Streams In (one of) the data streamng model(s), the data s a sequence of arrvals a 1, a 2,..., a m of the form a j = (, v) where s the dentty of the tem and belongs to
More informationDISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization
DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.
More informationMTH 819 Algebra I S13. Homework 1/ Solutions. 1 if p n b and p n+1 b 0 otherwise ) = 0 if p q or n m. W i = rw i
MTH 819 Algebra I S13 Homework 1/ Solutons Defnton A. Let R be PID and V a untary R-module. Let p be a prme n R and n Z +. Then d p,n (V) = dm R/Rp p n 1 Ann V (p n )/p n Ann V (p n+1 ) Note here that
More informationIntroductory Cardinality Theory Alan Kaylor Cline
Introductory Cardnalty Theory lan Kaylor Clne lthough by name the theory of set cardnalty may seem to be an offshoot of combnatorcs, the central nterest s actually nfnte sets. Combnatorcs deals wth fnte
More informationREDUCTION MODULO p. We will prove the reduction modulo p theorem in the general form as given by exercise 4.12, p. 143, of [1].
REDUCTION MODULO p. IAN KIMING We wll prove the reducton modulo p theorem n the general form as gven by exercse 4.12, p. 143, of [1]. We consder an ellptc curve E defned over Q and gven by a Weerstraß
More informationGraph Reconstruction by Permutations
Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer
More informationKuroda s class number relation
ACTA ARITMETICA XXXV (1979) Kurodas class number relaton by C. D. WALTER (Dubln) Kurodas class number relaton [5] may be derved easly from that of Brauer [2] by elmnatng a certan module of unts, but the
More informationOn cyclic of Steiner system (v); V=2,3,5,7,11,13
On cyclc of Stener system (v); V=,3,5,7,,3 Prof. Dr. Adl M. Ahmed Rana A. Ibraham Abstract: A stener system can be defned by the trple S(t,k,v), where every block B, (=,,,b) contans exactly K-elementes
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationLECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture.
LECTURE V EDWIN SPARK 1. More on the Chnese Remander Theorem We begn by recallng ths theorem, proven n the preceedng lecture. Theorem 1.1 (Chnese Remander Theorem). Let R be a rng wth deals I 1, I 2,...,
More informationLecture 4: Universal Hash Functions/Streaming Cont d
CSE 5: Desgn and Analyss of Algorthms I Sprng 06 Lecture 4: Unversal Hash Functons/Streamng Cont d Lecturer: Shayan Oves Gharan Aprl 6th Scrbe: Jacob Schreber Dsclamer: These notes have not been subjected
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationINVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS
INVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS HIROAKI ISHIDA Abstract We show that any (C ) n -nvarant stably complex structure on a topologcal torc manfold of dmenson 2n s ntegrable
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More informationErrata to Invariant Theory with Applications January 28, 2017
Invarant Theory wth Applcatons Jan Drasma and Don Gjswjt http: //www.wn.tue.nl/~jdrasma/teachng/nvtheory0910/lecturenotes12.pdf verson of 7 December 2009 Errata and addenda by Darj Grnberg The followng
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More information7. Products and matrix elements
7. Products and matrx elements 1 7. Products and matrx elements Based on the propertes of group representatons, a number of useful results can be derved. Consder a vector space V wth an nner product ψ
More informationGames of Threats. Elon Kohlberg Abraham Neyman. Working Paper
Games of Threats Elon Kohlberg Abraham Neyman Workng Paper 18-023 Games of Threats Elon Kohlberg Harvard Busness School Abraham Neyman The Hebrew Unversty of Jerusalem Workng Paper 18-023 Copyrght 2017
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, 2017 3 Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationCONJUGACY IN THOMPSON S GROUP F. 1. Introduction
CONJUGACY IN THOMPSON S GROUP F NICK GILL AND IAN SHORT Abstract. We complete the program begun by Brn and Squer of charactersng conjugacy n Thompson s group F usng the standard acton of F as a group of
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES. 1. Introduction
ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES TAKASHI ITOH AND MASARU NAGISA Abstract We descrbe the Haagerup tensor product l h l and the extended Haagerup tensor product l eh l n terms of
More informationThe Second Eigenvalue of Planar Graphs
Spectral Graph Theory Lecture 20 The Second Egenvalue of Planar Graphs Danel A. Spelman November 11, 2015 Dsclamer These notes are not necessarly an accurate representaton of what happened n class. The
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationMAT 578 Functional Analysis
MAT 578 Functonal Analyss John Qugg Fall 2008 Locally convex spaces revsed September 6, 2008 Ths secton establshes the fundamental propertes of locally convex spaces. Acknowledgment: although I wrote these
More informationSolutions for Tutorial 1
Toc 1: Sem-drect roducts Solutons for Tutoral 1 1. Show that the tetrahedral grou s somorhc to the sem-drect roduct of the Klen four grou and a cyclc grou of order three: T = K 4 (Z/3Z). 2. Show further
More informationk(k 1)(k 2)(p 2) 6(p d.
BLOCK-TRANSITIVE 3-DESIGNS WITH AFFINE AUTOMORPHISM GROUP Greg Gamble Let X = (Z p d where p s an odd prme and d N, and let B X, B = k. Then t was shown by Praeger that the set B = {B g g AGL d (p} s the
More informationTHERE ARE NO POINTS OF ORDER 11 ON ELLIPTIC CURVES OVER Q.
THERE ARE NO POINTS OF ORDER 11 ON ELLIPTIC CURVES OVER Q. IAN KIMING We shall prove the followng result from [2]: Theorem 1. (Bllng-Mahler, 1940, cf. [2]) An ellptc curve defned over Q does not have a
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More information= s j Ui U j. i, j, then s F(U) with s Ui F(U) G(U) F(V ) G(V )
1 Lecture 2 Recap Last tme we talked about presheaves and sheaves. Preshea: F on a topologcal space X, wth groups (resp. rngs, sets, etc.) F(U) or each open set U X, wth restrcton homs ρ UV : F(U) F(V
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationSUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION
talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776
More informationHMMT February 2016 February 20, 2016
HMMT February 016 February 0, 016 Combnatorcs 1. For postve ntegers n, let S n be the set of ntegers x such that n dstnct lnes, no three concurrent, can dvde a plane nto x regons (for example, S = {3,
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationBallot Paths Avoiding Depth Zero Patterns
Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,
More informationOn the smoothness and the totally strong properties for nearness frames
Int. Sc. Technol. J. Namba Vol 1, Issue 1, 2013 On the smoothness and the totally strong propertes for nearness frames Martn. M. Mugoch Department of Mathematcs, Unversty of Namba 340 Mandume Ndemufayo
More informationPartitions and compositions over finite fields
Parttons and compostons over fnte felds Muratovć-Rbć Department of Mathematcs Unversty of Saraevo Zmaa od Bosne 33-35, 71000 Saraevo, Bosna and Herzegovna amela@pmf.unsa.ba Qang Wang School of Mathematcs
More informationTHE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION
THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton
More informationSubset Topological Spaces and Kakutani s Theorem
MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered
More information