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1 Lecture 3. Group Actios PCMI Summer 2015 Udergraduate Lectures o Flag Varieties Lecture 3. The category of groups is discussed, ad the importat otio of a group actio is explored. Defiitio 3.1. A group is a set G with a compositio operatio (geerally but ot always writte as a product), such that: (i) Compositio is associative: f (g h) = (f g) h f, g, h. (ii) There is a elemet id G such that: g id = g = id g for all g. (iii) Every elemet of G has a (two-sided) iverse. Alteratively... It is also true (ad temptig) to defie: A mooid is a category with oe object, ad A group is the collectio of ivertible morphisms of a mooid Example 3.1. (a) Perm() or GL(V ), or ay of the automorphism groups Aut(X) of ay object X of ay category C. (b) A field k or vector space V with the additio operatio. (The group operatio i this case is additio. It s the exceptio provig the rule that the operatio is usually writte as a product.) (c) The uits k = k {0} of a field with the multiplicatio operatio. (d) The group with two elemets: {±1} with ( 1) ( 1) = 1. (e) The group {1} (aalogous to the empty set or the zero space). Defiitio 3.2. A map of groups φ : G G is a homomorphism if: (i) φ(id G ) = id G ad (ii) φ(g 1 g 2 ) = φ(g 1 ) φ(g 2 ) for all g 1, g 2 G. Example 3.2. (a) The (atural) logarithm is a group homomorphism l : R >0 R from the positive reals (with multiplicatio) to R (with additio). It is a isomorphism, with iverse the expoetial fuctio e x : R R >0. O the other had, the complex expoetial e z : C C is a group homomorphism that is oly locally ivertible by a logarithm. (b) The permutatio matrices of the previous lecture are the images of the homomorphism φ : Perm() GL(, k) give by φ(τ) = P τ. (c) The sig ad determiat are group homomorphisms: sg : Perm() {±1}, det : GL(, k) k 1

2 2 Momet of Ze. The category Groups of groups cosists of: (a) The collectio of all groups, ad (b) All group homomorphisms. Defiitio 3.3. A group G is abelia if the operatio is commutative. Example 3.3. Cosider the four elemet Klei group: K 4 := {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} Perm(4) This is a abelia group, with, for example: (1 2)(3 4) (1 3)(2 4) = (1 4)(2 3) = (1 3)(2 4) (1 2)(3 4) ad every elemet squares to the idetity. Defiitio 3.4. A abelia group G is cyclic if there is a elemet g G whose positive ad egative powers fill up G, i.e. {..., g 2, g 1, id, g, g 2, g 3,...} = G ad such a elemet g is said to geerate the cyclic group G. Example 3.4 (Cyclic Groups). (a) The itegers, Z, with additio, geerated by 1 (or 1). (b) The itegers (mod ), Z/Z, with additio, geerated by ay iteger (mod ) that is relatively prime to. (c) The ozero elemets, k, of a fiite field k, with multiplicatio, which is cyclic, but without a obvious choice of geerator! (d) The group of rotatioal symmetries of a regular polygo. Evidetly a cyclic group is abelia, but ot coversely: Products. Give groups G 1,..., G, the product G 1... G is the set of -tuples (g 1,..., g ) g i G i with coordiate-wise multiplicatio: (g 1,..., g ) (h 1,..., h ) = (g 1 h 1,..., g h ) ad id = (id,..., id) Defiitio 3.5. The order of a fiite group G is the umber G. Chiese Remaider. A product of fiite cyclic groups is cyclic if ad oly if the orders of the cyclic groups are pairwise relatively prime. Thus, if = k i=1 pm i i is the prime factorizatio of as a product of powers of distict primes, the: Z/Z = Z/p m 1 1 Z... Z/p m k k Example 3.5. The Klei group is ot cyclic (it is Z/2Z Z/2Z). Groups are far more complicated tha sets or vector spaces. For a idicatio of the complexity of groups, cosider the aalogue of the stadard sets [] ad the stadard vector spaces k.

3 Defiitio 3.6. The free group F () o geerators x 1,..., x is the set of words made up of letters x 1, x 1 1,..., x, x 1 (mod cacellatio) with cocateatio as the operatio (ad the empty word as the idetity). Remark. The group aalogue of Perm() or GL(, k), amely the group Aut(F ()) of automorphisms of F (), is extremely complicated. Defiitio 3.7. A group G is fiitely geerated if there is a surjective homomorphism: φ : F () G (for some value of ). A relatio is a equatio of two words that holds i G (after applyig φ). A set of relatios is complete if every equality of words i G (after applyig φ) is a cosequece of the give set of relatios. Example 3.6. (a) Cyclic groups are geerated by oe elemet x, with o o-trivial relatios i the ifiite case ad oe complete relatio (x = id) i the case of the cyclic group of order. (b) The dihedral group D 2 is geerated by x, y with relatios: (i) x = id, (ii) y 2 = id, (iii) yx = x 1 y There are 2 distict elemets of D 2 : {id, x, x 2,..., x 1, y, xy, x 2 y,..., x 1 y} Moreover, it follows that each x i y squares to zero, sice: (x i y) 2 = x i (yx i )y = x i (x i y)y = id Notice that if > 2, the the dihedral group D 2 is ot abelia. (c) Perm() is geerated by t 1 = (1 2), t 2 = (2 3),..., t 1 = ( 1 ). This takes some thought. For example: (1 3) = t 1 t 2 t 1 = t 2 t 1 t 2, (1 4) = t 1 t 2 t 3 t 2 t 1,..., ad a complete set of relatios is give by: t 2 i = id, t i t i+1 t i = t i+1 t i t i+1, t i t j = t j t i if i j > 1 (d) Droppig the relatios t 2 i = id i (c) but keepig the others defies the braid group o strads. This is a ifiite group which is fiitely geerated (with a complete set of fiitely may relatios). The most commoly studied groups fall, roughly, ito three types: (i) Fiite Groups (ii) Fiitely Geerated Ifiite Groups (e.g. F () ad braid groups) (iii) Cotiuous Groups (e.g. the real umbers with additio) We ll be iterested here i groups of type (i) ad (iii). Abelia groups, o the other had, are less complicated: 3

4 4 Theorem 3.1. Every fiitely geerated abelia group G is isomorphic to a product of cyclic groups: Z Z/d 1 Z... Z/d m Z for uique ad d 1,..., d m such that each d i divides d i+1. Still, the group GL(, Z) = Aut(Z ) is plety complicated. Defiitio 3.8. A homomorphism ρ : G Aut(X) is called a actio of the group G o a object X (of a category C). Examples iclude permutatio actios φ : G Aut(S) o a set ad liear actios, or represetatios ρ : G GL(V ) o a vector space. Example 3.7. (a) For each 0 m <, the map: χ m : C C = Aut(C 1 ); χ m (x) = e 2πim defies a oe-dimesioal complex represetatio of C. (b) The dihedral group D 2 has a two-dimesio real represetatio: 2π cos( ρ(x) = ) si( 2π) [ ] 1 0, ρ(y) = si( 2π) cos( 2π) 0 1 (rotatio couterclockwise by 2π/ ad reflectio across the x-axis). This is well-defied sice ρ(x) = id = ρ(y) 2 ad: 2π cos( ρ(y) ρ(x) = ) si( 2π) = ρ(x) 1 ρ(y) si( 2π) cos( 2π) (c) D 2 also acts o the set [] via: φ(x) = ( ), φ(y) = (1 1)(2 2) which is well-defied because φ(x) = id = φ(y) 2 ad φ(y)φ(x) = (1 2)(2 3)...( 1 ) = φ(x) 1 φ(y) This is the iduced actio from (b) o the vertices of a regular -go cetered at the origi, umbered couterclockwise, edig at = (1, 0). Defiitio 3.9. A subset H G is a subgroup if id H ad H is closed uder iverses ad the group operatios. Defiitio A actio φ : G Aut(S) o a set S is trasitive if, for each pair s, t S, there is a elemet g G such that φ(g)(s) = t, or equivaletly, if there is o proper (oempty) subset T S that is left fixed by the actio. The stabilizer H s of a elemet s S is the subgroup of elemets h G with the property that h(s) = s.

5 Propositio 3.1. If φ : G Aut(S) is a trasitive actio of a fiite group, ad if H s is the stabilizer of ay s S, the G = S H s. Proof. For each t S, choose g t G such that g t (s) = t. The: G = t S g t H s where g t H s = {g t h h H s } ad this is a disjoit uio of S sets, each of size H s. Corollary 3.1 (Lagrage s Theorem) If H G is a subgroup of a fiite group, the H divides G. Proof. Let S = {gh g G} be the set of left cosets of H G. The G acts o S by left multiplicatio: g (gh) = (g g)h, the actio is trasitive, ad the stabilizer of H S is H, so G = S H. I Propositio 2.3, the cosets of a subspace formed a vector space. Here, they are just a set (ad ot usually a group). But see below! Example 3.8. There are two obvious subgroups of D 2, amely: C = {1, x, x 2,..., x 1 } ad C 2 = {1, y} The cosets of C are: {1, x, x 2,..., x 1 } ad {y, yx,..., yx 1 }. The cosets of C 2 are {1, y}, {x, xy},..., {x 1, x 1 y}. Defiitio The cojugatio actio of G o itself is the map: c : G Aut(G), c(h)(g) = hgh 1 This is a actio i the category of groups sice: c(h)(gg ) = h(gg )h 1 = hgh 1 hg h 1 = c(h)(g) c(h)(g ) i.e. each c(h) is automorphism of G as a group. The image of the cojugatio actio is called the group of ier automorphisms of G. Example 3.9. (a) The cojugatio actio of a abelia group G o itself is trivial (i.e. the image of c : G Aut(G) is {id}), sice: c(h)(g) = hgh 1 = (hh 1 )g = g More geerally, if g G commutes with all other elemets of G, the g is said to be i the ceter of G, ad is fixed by the cojugatio actio. (b) The cojugatio actio of the permutatio group. If h Perm() (thought of as a bijectio h : [] []) ad g Perm() is writte i cycle otatio, the the cycle otatio for hgh 1 has the same shape as the cycle otatio for g, but with each i replaced with h(i). For example, if h = (1 2 3) Perm(3), the h(1 2)h 1 = (h(1) h(2)) = (2 3) ad h(1 3)h 1 = (2 1) = (1 2) 5

6 6 As a cosequece, it follows that the classes of elemets of Perm() are preserved by the cojugatio actio, ad the cojugatio actio is trasitive o each cojugacy class of elemets. For example, cosider the cojugacy classes of elemets i Perm(4): id, ( ), ( ), ( ), ( )( ) We ve see that these classes have 1, 6, 8, 6, 3 elemets, respectively. We ca write dow the stabilizers of elemets i each class: Stab(id) = Perm(4) Stab(1 2) = {id, (1 2), (1 2)(3 4), (3 4)} Stab(1 2 3) = {id, (1 2 3), (1 3 2)} Stab( ) = {id, ( ), (1 3)(2 4), ( )} Stab((1 2)(3 4)) = Exercise... ote that it has 24/3 = 8 elemets Let Sub(G) be the set of subgroups H G. Because it acts by group automorphisms, the cojugatio actio iduces a actio o the set of subgroups: φ : G Aut(Sub(G)); φ(g) = ghg 1 i.e. cojugatig a subgroup produces aother subgroup! Defiitio (a) Two subgroups H, H Sub(G) are cojugate if: ghg 1 = H for some g G i.e. if φ(g)(h) = H. The set of subgroups that are cojugate to H is called the cojugacy class of H. By defiitio, the, a group G acts trasitively o each cojugacy class of subgroups of G, just as it does o each cojugacy class of elemets of G. (b) A subgroup H G is ormal if it is fixed by cojugatio, i.e. if the cojugacy class of H cosists of H iself. Example The ormal subgroups of Perm(4) are {id}, K 4 ad: A 4 := {id, ( )( ), ( )} (12 elemets) i.e. all elemets σ Perm(4) such that sg(σ) = 1. As with vector spaces, we have the followig: Defiitio Let φ : G G be a group homomorphism. (i) The kerel of φ is the subgroup φ 1 (id G ) G. (ii) The image of φ is the subgroup im(φ) G. The followig is easily checked (see Propositio 2.2). Propositio 3.2. φ is ijective if ad oly if ker(φ) = {id G }.

7 However, there is a ew wrikle: Propositio 3.3. The kerel of a homomorphism φ : G G is a ormal subgroup of G ad coversely, if H G is a ormal subgroup, there is a surjective homomorphism q : G G with H = ker(φ). Proof. Suppose h ker(φ) ad h = ghg 1 for some g. The: φ(h ) = φ(ghg 1 ) = φ(g)φ(h)φ(g 1 ) = φ(g) id G φ(g) 1 = id G so h ker(φ). This implies that H is ormal. Coversely, suppose H G is ormal, ad cosider agai: G/H = {gh g G}, the set of cosets I claim that H beig a ormal subgroup is exactly the coditio that is eeded i order for multiplicatio of cosets to make G/H a group: (gh) (g H) = g(g Hg 1 )g H = (gg )H The the proof proceeds as i Propositio 2.3, with: q : G G/H give by q(g) = gh the surjective homomorphism with kerel H. Example The Klei subgroup K 4 Perm(4) is ormal, so it is the kerel of a group homomorphism. Lookig aroud, we fid it: φ : Perm(4) Aut({(1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}) amely, the cojugatio actio o the cojugacy class ( )( ). This is a surjective homomorphism to a group of order 6, with K 4 i the kerel (sice K 4 is commutative), hece K 4 is the kerel (4 6 = 24). Defiitio (a) The alteratig group Alt() Perm() is the kerel of the sig homomorphism (permutatios of sig 1). (b) The special liear group SL(, k) GL(, k) is the kerel of the determiat, i.e. the matrices A with det(a) = 1. Example Cosider Alt(5) with 60 = 5!/2 elemets i classes: id, ( ), ( )( ), ( ) of Perm(5) with 1, 20, 15, 24 elemets, respectively. This group has may subgroups but it is simple, meaig that it has o (o-trivial) ormal subgroups. Ideed, the groups Alt() for 5 are all simple. This is ot obvious, but you could prove it with the tools you have. Notice that the Perm(5) cojugacy class ( ) caot be a cojugacy class for Alt(5) actig o itself, sice 24 does ot divide 60. I fact, it splits i two. 7

8 8 Exercises. 1. Show that a actio of a group G o a set S is the same as a map from the Cartesia product: a : G S S; writte a(g, s) = gs with the property that g 1 (g 2 s) = (g 1 g 2 )s for all g 1, g 2 G. If S = V is a vector space over k, the the actio is a represetatio if i additio, multiplicatio by g is liear, i.e. g( v + w) = g v +g w ad g(c v) = cg( v). 2. If G acts trasitively o S ad H s ad H s are stabilizers of s, s S, prove that H s ad H s are cojugate subgroups of G. Coversely, if H = gh s g 1 is cojugate to H s, for some g G, prove that H = H s for some s S. Thus, S idexes the elemets i the cojugacy class of the stabilizer of (ay) elemet s S. 3. Prove that the left multiplicatio actio of G o itself is trasitive, but is oly a automorphism of G as a set, ad ot a actio i the category of groups (i cotrast to the cojugatio actio). 4. Prove that the group of uits (deoted k ) of a fiite field is cyclic. Hit: If = k, the every elemet of k is a root of x 1. If g k is ot a geerator, the g is a root of x d 1 for some d that properly divides (by Lagrage s Theorem). The problem ow follows from coutig ad the fact that a polyomial of degree d with coefficiets i a field has o more tha d roots i that field. 5. Prove the Chiese Remaider result. 6. For the dihedral groups D 2 : (a) Fid all the cojugacy classes of elemets g D 2. (b) Fid all the ormal subgroups H D Fid all cojugacy classes of elemets of (a) Alt(4) (b) Alt(5). 8. Fid the stabilizer of (1 2)(3 4) {(1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} for the cojugatio actio of Perm(4). (It is a group with 8 elemets). 9. Fid Aut(G) (i the category of groups) for the groups C ad D Idetify the groups of rotatioal symmetries of: (a) A regular tetrahedro. (b) A cube. (c) A regular dodecahedro. Hit: Two are alteratig groups ad oe is a permutatio group. Ze master problems.

9 11. Show that a product of two groups as defied just above Defiitio 3.5 is a product i the category of groups. A fuctor F : C D is faithful if F is ijective o objects ad each ( ) F : Hom(X, Y ) Hom(F (X), F (Y )) is also ijective. I other words, each object ad arrow from C has a uique image i D. It is fully faithful if each of the maps ( ) is also surjective (but F is still oly required to be ijective o objects). 12. Show that the forgetful fuctors from vector spaces over k to sets, as well as the forgetful fuctor from groups to sets, are all faithful but ot fully faithful. Notice that all groups ad vector spaces are poited, i.e. they have a distiguished elemets, so forgetful fuctors to the category of sets pass through a category whose objects cosist of sets with a distiguished poit (what are the morphisms?). However, eve the fuctor from the category of sets with a distiguished poit to sets (forgettig the poit) is ot fully faithful. 13. Let Ab be the category of abelia groups. Show that the forgetful fuctors from vector spaces over ay k to Ab are all faithful, but ot fully faithful, but i cotrast, show that the fuctor from abelia groups to groups is fully faithful. 9