Chapter 8: Homomorphisms

Transcription

1 Chapt 8: Homomophisms Matthw Macauly Dpatmnt of Mathmatical Scincs Clmson Univsity Math 42, Summ I 24 M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 / 5

2 Ovviw Thoughout th cous, w v said things lik: This goup has th sam stuctu as that goup. This goup is isomophic to that goup. Howv, w v nv ally splld out th dtails about what this mans. W will study a spcial typ of function btwn goups, calld a homomophism. An isomophism is a spcial typ of homomophism. Th Gk oots homo and moph togth man sam shap. Th a two situations wh homomophisms ais: whn on goup is a subgoup of anoth; whn on goup is a quotint of anoth. Th cosponding homomophisms a calld mbddings and quotint maps. Also in this chapt, w will compltly classify all finit ablian goups, and gt a tast of a fw mo advancd topics, such as th th fou isomophism thoms, commutatos subgoups, and automophisms. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 2 / 5

3 A motivating xampl Consid th statmnt: Z 3 < D 3. H is a visual: f f f 2 2 Th goup D 3 contains a siz-3 cyclic subgoup, which is idntical to Z 3 in stuctu only. Non of th lmnts of Z 3 (namly,, 2) a actually in D 3. Whn w say Z 3 < D 3, w ally man that th stuctu of Z 3 shows up in D 3. In paticula, th is a bijctiv cospondnc btwn th lmnts in Z 3 and thos in th subgoup in D 3. Futhmo, th lationship btwn th cosponding nods is th sam. A homomophism is th mathmatical tool fo succinctly xpssing pcis stuctual cospondncs. It is a function btwn goups satisfying a fw natual poptis. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 3 / 5

4 Homomophisms Using ou pvious xampl, w say that this function maps lmnts of Z 3 to lmnts of D 3. W may wit this as 2 f φ(n) = n φ: Z 3 D f f Th goup fom which a function oiginats is th domain (Z 3 in ou xampl). Th goup into which th function maps is th codomain (D 3 in ou xampl). Th lmnts in th codomain that th function maps to a calld th imag of th function ({,, 2 } in ou xampl), dnotd Im(φ). That is, Dfinition Im(φ) = φ(g) = {φ(g) g G}. A homomophism is a function φ: G H btwn two goups satisfying φ(ab) = φ(a)φ(b), fo all a, b G. Not that th opation a b is occuing in th domain whil φ(a) φ(b) occus in th codomain. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 4 / 5

5 Homomophisms Rmak Not vy function fom on goup to anoth is a homomophism! Th condition φ(ab) = φ(a)φ(b) mans that th map φ psvs th stuctu of G. Th φ(ab) = φ(a)φ(b) condition has visual intptations on th lvl of Cayly diagams and multiplication tabls. Cayly diagams a ab = c Domain b c φ Codomain φ(a) φ(b) φ(a)φ(b) = φ(c) φ(c) b φ(b) Multiplication tabls a c φ φ(a) φ(c) Not that in th Cayly diagams, b and φ(b) a paths; thy nd not just b dgs. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 5 / 5

6 An xampl Consid th function φ that ducs an intg modulo 5: φ: Z Z 5, φ(n) = n (mod 5). Sinc th goup opation is additiv, th homomophism popty bcoms φ(a + b) = φ(a) + φ(b). In plain English, this just says that on can fist add and thn duc modulo 5, OR fist duc modulo 5 and thn add. Cayly diagams 9 Domain: Z 8 φ 4 Codomain: Z Addition tabls 9 27 φ 4 2 M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 6 / 5

7 Typs of homomophisms Consid th following homomophism θ : Z 3 C 6, dfind by θ(n) = 2n : It is asy to chck that θ(a + b) = θ(a)θ(b): Th d-aow in Z 3 (psnting ) gts mappd to th 2-stp path psnting 2 in C 6. A homomophism φ: G H that is on-to-on o injctiv is calld an mbdding: th goup G mbds into H as a subgoup. If θ is not on-to-on, thn it is a quotint. If φ(g) = H, thn φ is onto, o sujctiv. Dfinition A homomophism that is both injctiv and sujctiv is an isomophism. An automophism is an isomophism fom a goup to itslf. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 7 / 5

8 Homomophisms and gnatos Rmak If w know wh a homomophism maps th gnatos of G, w can dtmin wh it maps all lmnts of G. Fo xampl, suppos φ : Z 3 Z 6 was a homomophism, with φ() = 4. Using this infomation, w can constuct th st of φ: Exampl φ(2) = φ( + ) = φ() + φ() = = 2 φ() = φ( + 2) = φ() + φ(2) = =. Suppos that G = a, b, and φ: G H, and w know φ(a) and φ(b). Using this infomation w can dtmin th imag of any lmnt in G. Fo xampl, fo g = a 3 b 2 ab, w hav φ(g) = φ(aaabbab) = φ(a) φ(a) φ(a) φ(b) φ(b) φ(a) φ(b). What do you think φ(a ) is? M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 8 / 5

9 Two basic poptis of homomophisms Poposition Lt φ: G H b a homomophism. Dnot th idntity of G by G, and th idntity of H by H. (i) φ( G ) = H (ii) φ(g ) = φ(g) φ snds th idntity to th idntity φ snds invss to invss Poof (i) Pick any g G. Now, φ(g) H; obsv that φ( G ) φ(g) = φ( G g) = φ(g) = H φ(g). Thfo, φ( G ) = H. (ii) Tak any g G. Obsv that φ(g) φ(g ) = φ(gg ) = φ( G ) = H. Sinc φ(g)φ(g ) = H, it follows immdiatly that φ(g ) = φ(g). M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 9 / 5

10 A wod of caution Just bcaus a homomophism φ: G H is dtmind by th imag of its gnatos dos not man that vy such imag will wok. Fo xampl, suppos w ty to dfin a homomophism φ: Z 3 Z 4 by φ() =. Thn w gt φ(2) = φ( + ) = φ() + φ() = 2, φ() = φ( + + ) = φ() + φ() + φ() = 3. This is impossibl, bcaus φ() =. (Idntity is mappd to th idntity.) That s not to say that th isn t a homomophism φ: Z 3 Z 4; not that th is always th tivial homomophism btwn two goups: φ: G H, φ(g) = H fo all g G. Excis Show that th is no mbdding φ: Z n Z, fo n 2. That is, any such homomophism must satisfy φ() =. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 / 5

11 Isomophisms Two isomophic goups may nam thi lmnts diffntly and may look diffnt basd on th layouts o choic of gnatos fo thi Cayly diagams, but th isomophism btwn thm guaants that thy hav th sam stuctu. Whn two goups G and H hav an isomophism btwn thm, w say that G and H a isomophic, and wit G = H. Th oots of th polynomial f (x) = x 4 a calld th 4th oots of unity, and dnotd R(4) := {, i,, i}. Thy a a subgoup of C := C \ {}, th nonzo complx numbs und multiplication. Th following map is an isomophism btwn Z 4 and R(4). φ: Z 4 R(4), φ(k) = i k. 3 2 i i M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 / 5

12 Isomophisms Somtims, th isomophism is lss visually obvious bcaus th Cayly gaphs hav diffnt stuctu. Fo xampl, th following is an isomophism: φ: Z 6 C φ(k) = k H is anoth non-obvious isomophism btwn S 3 = (2), (23) and D 3 =, f. f f φ: S 3 D 3 φ: (2) 2 f φ: (23) f (23) (2) f 2 f (32) (32) 2 f (3) M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 2 / 5

13 Anoth xampl: th quatnions Lt GL n(r) b th st of invtibl n n matics with al-valud ntis. It is asy to s that this is a goup und multiplication. Rcall th quatnion goup Q 4 = i, j, k i 2 = j 2 = k 2 =, ij = k. Th following st of 8 matics foms an isomophic goup und multiplication, wh I is th 4 4 idntity matix: { ±I, ± [ ], ± [ ], ± Fomally, w hav an mbdding φ: Q 4 GL 4(R) wh φ(i) = [ ], φ(j) = [ ], φ(k) = W say that Q 4 is psntd by a st of matics. [ ]}. [ ]. Many oth goups can b psntd by matics. Can you think of how to psnt V 4, C n, o S n, using matics? M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 3 / 5

14 Quotint maps Consid a homomophism wh mo than on lmnt of th domain maps to th sam lmnt of codomain (i.., non-mbddings). H a som xampls. τ : Q 4 V 4 τ 2 : Z Z 6 2 i i h j j k k v Non-mbdding homomophisms a calld quotint maps (as w ll s, thy cospond to ou quotint pocss). M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 4 / 5

15 Pimags Dfinition If φ: G H is a homomophism and h Im(φ) < H, dfin th pimag of h to b th st φ (h) := {g G : φ(g) = h}. Obsv in th pvious xampls that th pimags all had th sam stuctu. This always happns. Domain A a p a 2 φ Codomain a B b p b 2. b. Th pimag of H H is calld th knl of φ, dnotd K φ. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 5 / 5

16 Pimags Obsvation All pimags of φ hav th sam stuctu. Poof (sktch) Pick two lmnts a, b φ(g), and lt A = φ (a) and B = φ (b) b thi pimags. p Consid any path a a 2 btwn lmnts in A. Fo any b B, th is a p cosponding path b b 2. W nd to show that b 2 B. Sinc homomophisms psv stuctu, φ(a ) φ(p) φ(a 2). Sinc φ(a ) = φ(a 2), φ(p) is th mpty path. Thfo, φ(b ) φ(p) φ(b 2), i.., φ(b ) = φ(b 2), and so by dfinition, b 2 B. Claly, G is patitiond by pimags of φ. Additionally, w just showd that thy all hav th sam stuctu. (Sound familia?) M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 6 / 5

17 Pimags and knls Dfinition Th knl of a homomophism φ: G H is th st K(φ) := φ () = {k G : φ(k) = }. Obsvation 2 (i) Th pimag of th idntity (i.., K = K(φ)) is a subgoup of G. (ii) All oth pimags a lft costs of K. Poof (of (i)) Lt K = K(φ), and tak a, b K. W must show that K satisfis 3 poptis: Idntity: φ() =. Closu: φ(ab) = φ(a) φ(b) = =. Invss: φ(a ) = φ(a) = =. Thus, K is a subgoup of G. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 7 / 5

18 Knls Obsvation 3 K(φ) is a nomal subgoup of G. Poof Lt K = K(φ). W will show that if k K, thn gkg K. Tak any g G, and obsv that φ(gkg ) = φ(g) φ(k) φ(g ) = φ(g) φ(g ) = φ(g)φ(g) =. Thfo, gkg K(φ), so K G. Ky obsvation Givn any homomophism φ: G H, w can always fom th quotint goup G/ K(φ). M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 8 / 5

19 Quotints: via multiplication tabls Rcall that C 2 = { πi, πi } = {, }. Consid th following (quotint) homomophism: φ: D 4 C 2, dfind by φ() = and φ(f ) =. Not that φ(otation) = and φ(flction) =. Th quotint pocss of shinking D 4 down to C 2 can b claly sn fom th multiplication tabls. 2 3 f f 2 3 f f 2 f 3 f f 2 f f f 3 f 2 f f f f f f 3 f f 3 f 2 f f f f 3 f 2 f 2 3 f f f 3 f f 2 f 3 f f 2 f 3 f f f f f f f 2 f 3 f 2 3 f f 2 f 3 f 2 3 f 2 f 3 f f non-flip flip f 3 f f f f f f 2 f f f 3 f 2 f f 3 2 f f f 3 f 2 f 3 flip non-flip 2 2 f 2 f f f 3 f f 3 f 2 f f f 3 2 M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 9 / 5

20 Quotints: via Cayly diagams Dfin th homomophism φ : Q 4 V 4 via φ(i) = v and φ(j) = h. Sinc Q 4 = i, j, w can dtmin wh φ snds th maining lmnts: φ() =, φ( ) = φ(i 2 ) = φ(i) 2 = v 2 =, φ(k) = φ(ij) = φ(i)φ(j) = vh =, φ( k) = φ(ji) = φ(j)φ(i) = hv =, φ( i) = φ( )φ(i) = v = v, φ( j) = φ( )φ(j) = h = h. Not that K φ = {, }. Lt s s what happns whn w quotint out by K φ: i i K i i ik K ik Q 4 Q 4 Q 4/K j j k Q 4 oganizd by th subgoup K = k jk j j k lft costs of K a na ach oth Do you notic any lationship btwn Q 4/ K(φ) and Im(φ)? k kk jk kk collaps costs into singl nods M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 2 / 5

21 Th Fundamntal Homomophism Thom Th following sult is on of th cowning achivmnts of goup thoy. Fundamntal homomophism thom (FHT) If φ: G H is a homomophism, thn Im(φ) = G/ K(φ). Th FHT says that vy homomophism can b dcomposd into two stps: (i) quotint out by th knl, and thn (ii) labl th nods via φ. G (K φ G) φ any homomophism Im φ quotint pocss q i maining isomophism ( labling ) G / K φ goup of costs M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 2 / 5

22 Poof of th FHT Fundamntal homomophism thom If φ: G H is a homomophism, thn Im(φ) = G/ K(φ). Poof W will constuct an xplicit map i : G/ K(φ) Im(φ) and pov that it is an isomophism. Lt K = K(φ), and call that G/K = {ak : a G}. Dfin i : G/K Im(φ), i : gk φ(g). Show i is wll-dfind : W must show that if ak = bk, thn i(ak) = i(bk). Suppos ak = bk. W hav By dfinition of b a K(φ), ak = bk = b ak = K = b a K. H = φ(b a) = φ(b ) φ(a) = φ(b) φ(a) = φ(a) = φ(b). By dfinition of i: i(ak) = φ(a) = φ(b) = i(bk). M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

23 Poof of FHT (cont.) [Rcall: i : G/K Im(φ), i : gk φ(g)] Poof (cont.) Show i is a homomophism : W must show that i(ak bk) = i(ak) i(bk). i(ak bk) = i(abk) (ak bk := abk) = φ(ab) (dfinition of i) = φ(a) φ(b) (φ is a homomophism) = i(ak) i(bk) (dfinition of i) Thus, i is a homomophism. Show i is sujctiv (onto) : This mans showing that fo any lmnt in th codomain (h, Im(φ)), that som lmnt in th domain (h, G/K) gts mappd to it by i. Pick any φ(a) Im(φ). By dfintion, i(ak) = φ(a), hnc i is sujctiv. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

24 Poof of FHT (cont.) [Rcall: i : G/K Im(φ), i : gk φ(g)] Poof (cont.) Show i is injctiv ( ) : W must show that i(ak) = i(bk) implis ak = bk. Suppos that i(ak) = i(bk). Thn i(ak) = i(bk) = φ(a) = φ(b) (by dfinition) = φ(b) φ(a) = H = φ(b a) = H (φ is a homom.) = b a K (dfinition of K(φ)) = b ak = K (ah = H a H) = ak = bk Thus, i is injctiv. In summay, sinc i : G/K Im(φ) is a wll-dfind homomophism that is injctiv ( ) and sujctiv (onto), it is an isomophism. Thfo, G/K = Im(φ), and th FHT is povn. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

25 Consquncs of th FHT Coollay If φ: G H is a homomophism, thn Im φ H. A fw spcial cass If φ: G H is an mbdding, thn K(φ) = { G }. Th FHT says that Im(φ) = G/{ G } = G. If φ: G H is th map φ(g) = H fo all h G, thn K(φ) = G, so th FHT says that { H } = Im(φ) = G/G. Lt s us th FHT to dtmin all homomophisms φ: C 4 C 3: By th FHT, G/ K φ = Im φ < C 3, and so Im φ = o 3. Sinc K φ < C 4, Lagang s Thom also tlls us that K φ {, 2, 4}, and hnc Im φ = G/ K φ {, 2, 4}. Thus, Im φ =, and so th only homomophism φ: C 4 C 3 is th tivial on. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

26 How to show two goups a isomophic Th standad way to show G = H is to constuct an isomophism φ: G H. Whn th domain is a quotint, th is anoth mthod, du to th FHT. Usful tchniqu Suppos w want to show that G/N = H. Th a two appoachs: (i) Dfin a map φ: G/N H and pov that it is wll-dfind, a homomophism, and a bijction. (ii) Dfin a map φ: G H and pov that it is a homomophism, a sujction (onto), and that K φ = N. Usually, Mthod (ii) is asi. Showing wll-dfindnss and injctivity can b ticky. Fo xampl, ach of th following a sults that w will s vy soon, fo which (ii) woks quit wll: Z/ n = Z n; Q / = Q + ; AB/B = A/(A B) G/(A B) = (G/A) (G/B) (assuming A, B G); (assuming G = AB). M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

27 Cyclic goups as quotints Consid th following nomal subgoup of Z: 2Z = 2 = {..., 24, 2,, 2, 24,... } Z. Th lmnts of th quotint goup Z/ 2 a th costs: + 2, + 2, 2 + 2,..., + 2, + 2. Numb thoists call ths sts congunc classs mod 2. W say that two numbs a congunt mod 2 if thy a in th sam cost. Rcall how to add costs in th quotint goup: (a + 2 ) + (b + 2 ) := (a + b) + 2. (Th cost containing a) + (th cost containing b) = th cost containing a + b. It should b cla that Z/ 2 is isomophic to Z 2. Fomally, this is just th FHT applid to th following homomophism: φ: Z Z 2, φ: k k (mod 2), Claly, K(φ) = {..., 24, 2,, 2, 24,... } = 2. By th FHT: Z/ K(φ) = Z/ 2 = Im(φ) = Z 2. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

28 A pictu of th isomophism i : Z 2 Z/ 2 (fom th VGT wbsit) M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

29 Finit ablian goups W v sn that som cyclic goups can b xpssd as a dict poduct, and oths cannot. Blow a two ways to lay out th Cayly diagam of Z 6 so th dict poduct stuctu is obvious: Z 6 = Z3 Z Howv, th goup Z 8 cannot b wittn as a dict poduct. No matt how w daw th Cayly gaph, th must b an lmnt (aow) of od 8. Why? W will answ th qustion of whn Z n Z m = Znm, and in doing so, compltly classify all finit ablian goups. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

30 Finit ablian goups Poposition Z nm = Zn Z m if and only if gcd(n, m) =. Poof (sktch) : Suppos gcd(n, m) =. W claim that (, ) Z n Z m has od nm. (, ) is th smallst k such that (k, k) = (, ). This happns iff n k and m k. Thus, k = lcm(n, m) = nm. (,) (,) (2,) (3,) Th following imag illustats this using th Cayly diagam in th goup Z 4 Z 3 = Z2. (,) (,) (2,) (3,) (,2) (,2) (2,2) (3,2) M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 3 / 5

31 Finit ablian goups Poposition Z nm = Zn Z m if an only if gcd(n, m) =. Poof (cont.) : Suppos Z nm = Zn Z m. Thn Z n Z m has an lmnt (a, b) of od nm. Fo convninc, w will switch to multiplicativ notation, and dnot ou cyclic goups by C n. Claly, a = C n and b = C m. Lt s look at a Cayly diagam fo C n C m. Th od of (a, b) must b a multipl of n (th numb of ows), and of m (th numb of columns). By dfinition, this is th last common multipl of n and m. (,) (,b)... (,b m- ) (a,) (a,b)... (a,b m- ) (a n-,) (a n-,b)... a n-,b m- But (a, b) = nm, and so lcm(n, m) = nm. Thfo, gcd(n, m) =. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 3 / 5

32 Th Fundamntal Thom of Finit Ablian Goups Classification thom (by pim pows ) Evy finit ablian goup A is isomophic to a dict poduct of cyclic goups, i.., fo som intgs n, n 2,..., n m, A = Z n Z n2 Z nm, wh ach n i is a pim pow, i.., n i = p d i i, wh p i is pim and d i N. Th poof of this is mo advancd, and whil it is at th undgaduat lvl, w don t yt hav th tools to do it. Howv, w will b mo intstd in undstanding and utilizing this sult. Exampl Up to isomophism, th a 6 ablian goups of od 2 = : Z 8 Z 25 Z 8 Z 5 Z 5 Z 2 Z 4 Z 25 Z 2 Z 4 Z 5 Z 5 Z 2 Z 2 Z 2 Z 25 Z 2 Z 2 Z 2 Z 5 Z 5 M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

33 Th Fundamntal Thom of Finit Ablian Goups Finit ablian goups can b classifid by thi lmntay divisos. Th mystious tminology coms fom th thoy of moduls (a gaduat-lvl topic). Classification thom (by lmntay divisos ) Evy finit ablian goup A is isomophic to a dict poduct of cyclic goups, i.., fo som intgs k, k 2,..., k m, wh ach k i is a multipl of k i+. A = Z k Z k2 Z km. Exampl Up to isomophism, th a 6 ablian goups of od 2 = : by pim-pows by lmntay divisos Z 8 Z 25 Z 2 Z 4 Z 2 Z 25 Z Z 2 Z 2 Z 2 Z 2 Z 25 Z 5 Z 2 Z 2 Z 8 Z 5 Z 5 Z 4 Z 5 Z 4 Z 2 Z 5 Z 5 Z 2 Z Z 2 Z 2 Z 2 Z 5 Z 5 Z Z Z 2 M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

34 Th Fundamntal Thom of Finitly Gnatd Ablian Goups Just fo fun, h is th classification thom fo all finitly gnatd ablian goups. Not that it is not much diffnt. Thom Evy finitly gnatd ablian goup A is isomophic to a dict poduct of cyclic goups, i.., fo som intgs n, n 2,..., n m, A = Z Z Z }{{} n Z n2 Z nm, k copis wh ach n i is a pim pow, i.., n i = p d i i, wh p i is pim and d i N. In oth wods, A has th following goup psntation: A = a,..., a k,,..., m n = = nm m =. In summay, ablian goups a lativly asy to undstand. In contast, nonablian goups a mo mystious and complicatd. Soon, w will study th Sylow Thoms which will hlp us btt undstand th stuctu of finit nonablian goups. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

35 Th Isomophism Thoms Th Fundamntal Homomophism Thom (FHT) is th fist of fou basic thoms about homomophism and thi stuctu. Ths a commonly calld Th Isomophism Thoms : Fist Isomophism Thom: Fundamntal Homomophism Thom Scond Isomophism Thom: Diamond Isomophism Thom Thid Isomophism Thom: Fshman Thom Fouth Isomophism Thom: Cospondnc Thom All of ths thoms hav analogus in oth algbaic stuctus: ings, vcto spacs, moduls, and Li algbas, to nam a fw. In th maind of this chapt, w will summaiz th last th isomophism thoms, and povid visual pictus fo ach. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

36 Th Scond Isomophism Thom Diamond isomophism thom Lt H G, and N G. Thn (i) Th poduct HN = {hn h H, n N} is a subgoup of G. (ii) Th intsction H N is a nomal subgoup of G. (iii) Th following quotint goups a isomophic: HN/N = H/(H N) G HN H N H N Poof (sktch) Dfin th following map φ: H HN/N, φ: h hn. If w can show:. φ is a homomophism, 2. φ is sujctiv (onto), 3. K φ = H N, thn th sult will follow immdiatly fom th FHT. Th dtails a lft as HW. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

37 Th Thid Isomophism Thom Fshman thom Consid a chain N H G of nomal subgoups of G. Thn. Th quotint H/N is a nomal subgoup of G/N; 2. Th following quotints a isomophic: (G/N)/(H/N) = G/H. G G/N H N H/N (G/N) (H/N) = G H (Thanks to Zach Titl of Bois Stat fo th concpt and gaphic!) M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

38 Th Thid Isomophism Thom Fshman thom Consid a chain N H G of nomal subgoups of G. Thn H/N G/N and (G/N)/(H/N) = G/H. Poof It is asy to show that H/N G/N (xcis). Dfin th map ϕ: G/N G/H, ϕ: gn gh. Show ϕ is wll-dfind : Suppos g N = g 2N. Thn g = g 2n fo som n N. But n H bcaus N H. Thus, g H = g 2H, i.., ϕ(g N) = ϕ(g 2N). ϕ is claly onto and a homomophism. Apply th FHT: K ϕ = {gn G/N ϕ(gn) = H} = {gn G/N gh = H} = {gn G/N g H} = H/N By th FHT, (G/N)/ K ϕ = (G/N)/(H/N) = Im ϕ = G/H. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

39 Th Fouth Isomophism Thom Th full statmnt is a bit tchnical, so h w just stat it infomally. Cospondnc thom Lt N G. Th is a cospondnc btwn subgoups of G/N and subgoups of G that contain N. In paticula, vy subgoup of G/N has th fom H/N fo som H satisfying N H G. This mans that th cosponding subgoup lattics a idntical in stuctu. Exampl Q 4 Q 4 / V 4 i j k i / j / k / h vh v / Th quotint Q 4/ is isomophic to V 4. Th subgoup lattics can b visualizd by collapsing to th idntity. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

40 Cospondnc thom (fomally) Lt N G. Thn th is a bijction fom th subgoups of G/N and subgoups of G that contain N. In paticula, vy subgoup of G/N has th fom A := A/N fo som A satisfying N A G. Moov, if A, B G, thn. A B if and only if A B; 2. If A B, thn [B : A] = [B : A]; 3. A, B = A, B, 4. A B = A B, 5. A G if and only if A G Exampl D 4 D 4 / 2 V 4 2, f 2, f 2, f / 2 / 2 2, f / 2 h vh v f 2 f 2 f 3 f 2 / 2 M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 4 / 5

41 Commutato subgoups and ablianizations W v sn how to divid Z by 2, thby focing all multipls of 2 to b zo. This is on way to constuct th intgs modulo 2: Z 2 = Z/ 2. Now, suppos G is nonablian. W would lik to divid G by its non-ablian pats, making thm zo and laving only ablian pats in th sulting quotint. A commutato is an lmnt of th fom aba b. Sinc G is nonablian, th a non-idntity commutatos: aba b in G. ab = ba ab ba In this cas, th st C := {aba b a, b G} contains mo than th idntity. Dfin th commutato subgoup G of G to b G := aba b a, b G. This is a nomal subgoup of G (homwok xcis), and if w quotint out by it, w gt an ablian goup! (Bcaus w hav killd vy instanc of th ab ba pattn shown abov.) M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 4 / 5

42 Commutato subgoups and ablianizations Dfinition Th ablianization of G is th quotint goup G/G. This is th goup that on gts by killing off all nonablian pats of G. In som sns, th commutato subgoup G is th smallst nomal subgoup N of G such that G/N is ablian. [Not that G would b th lagst such subgoup.] Equivalntly, th quotint G/G is th lagst ablian quotint of G. [Not that G/G = would b th smallst such quotint.] Univsal popty of commutato subgoups Suppos f : G A is a homomophism to an ablian goup A. Thn th is a uniqu homomophism h : G/G A such that f = hq: f G A q h G/G W say that f factos though th ablianization, G/G. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

43 Commutato subgoups and ablianizations Exampls Consid th goups A 4 and D 4. It is asy to chck that G A 4 = xyx y x, y A 4 = V 4, G D 4 = xyx y x, y D 4 = 2. A 4 D 4 (2)(34), (3)(24) 2, f 2, f (23) (24) (34) (234) (2)(34)) (3)(24) (4)(23)) f 2 f 2 f 3 f {} Thus, th ablianization of A 4 is A 4/V 4 = C3, and th ablianization of D 4 is D 4/ 2 = V 4. Notic that G/G is ablian, and moov, taking th quotint of G by anything abov G will yild an ablian goup. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

44 Automophisms Dfinition An automophism is an isomophism fom a goup to itslf. Th st of all automophisms of G foms a goup, calld th automophism goup of G, and dnotd Aut(G). Rmaks. An automophism is dtmind by wh it snds th gnatos. An automophism φ must snd gnatos to gnatos. In paticula, if G is cyclic, thn it dtmins a pmutation of th st of (all possibl) gnatos. Exampls. Th a two automophisms of Z: th idntity, and th mapping n n. Thus, Aut(Z) = C Th is an automophism φ: Z 5 Z 5 fo ach choic of φ() {, 2, 3, 4}. Thus, Aut(Z 5) = C 4 o V 4. (Which on?) 3. An automophism φ of V 4 = h, v is dtmind by th imag of h and v. Th a 3 choics fo φ(h), and thn 2 choics fo φ(v). Thus, Aut(V 4) = 6, so it is ith C 6 = C2 C 3, o S 3. (Which on?) M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

45 Automophism goups of Z n Dfinition Th multiplicativ goup of intgs modulo n, dnotd Z n o U(n), is th goup U(n) := {k Z n gcd(n, k) = } wh th binay opation is multiplication, modulo n. U(5) = {, 2, 3, 4} = C 4 U(8) = {, 3, 5, 7} = C 2 C U(6) = {, 5} = C Poposition (homwok) Th automophism goup of Z n is Aut(Z n) = {σ a a U(n)} = U(n), wh σ a : Z n Z n, σ a() = a. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

46 Automophisms of D 3 Lt s find all automophisms of D 3 =, f. W ll s a vy simila xampl to this whn w study Galois thoy. Claly, vy automophism φ is compltly dtmind by φ() and φ(f ). Sinc automophisms psv od, if φ Aut(D 3), thn φ() =, φ() = } o {{ } 2, φ(f ) = f, f, o 2 f. }{{} 2 choics 3 choics Thus, th a at most 2 3 = 6 automophisms of D 3. Lt s ty to dfin two maps, (i) α: D 3 D 3 fixing, and (ii) β : D 3 D 3 fixing f : { α() = α(f ) = f { β() = 2 β(f ) = f I claim that: ths both dfin automophisms (chck this!) ths gnat six diffnt automophisms, and thus α, β = Aut(D 3). To dtmin what goup this is isomophic to, find ths six automophisms, and mak a goup psntation and/o multiplication tabl. Is it ablian? M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

47 Automophisms of D 3 An automophism can b thought of as a -wiing of th Cayly diagam. id f f 2 2 f f f f f 2 f f f f f f 2 f 2 β 2 f f α f f 2 2 f f f f f 2 f f f f f f 2 f 2 αβ 2 f 2 f α 2 f 2 f 2 2 f f f f f 2 f f f f f f 2 f 2 α2 β 2 f f M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

48 Automophisms of D 3 H is th multiplication tabl and Cayly diagam of Aut(D 3) = α, β. id id α α 2 β αβ α 2 β id α α 2 β αβ α 2 β id α α 2 α α 2 α 2 β αβ α 2 id αβ α 2 β β id α α 2 β β αβ β α 2 β αβ id αβ β α 2 β α β β αβ α 2 β α 2 α 2 id α α α 2 id It is puly coincidnc that Aut(D 3) = D 3. Fo xampl, w v alady sn that Aut(Z 5) = U(5) = C 4, Aut(Z 6) = U(6) = C 2, Aut(Z 8) = U(8) = C 2 C 2. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

49 Automophisms of V 4 = h, v Th following pmutations a both automophisms: α : h v hv and β : h v hv h id h h h β v h v v v h hv hv v hv hv hv v hv h α v h h αβ h h v hv v hv hv h v hv hv v v hv h α2 hv v h h h α2 β hv v v h hv v v hv hv h v hv M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I / 5

50 Automophisms of V 4 = h, v H is th multiplication tabl and Cayly diagam of Aut(V 4) = α, β = S 3 = D3. id id α α 2 β αβ α 2 β id α α 2 β αβ α 2 β id α α 2 α α 2 α 2 β αβ α 2 id αβ α 2 β β id α α 2 β β αβ β α 2 β αβ id αβ β α 2 β α β β αβ α 2 β α 2 α 2 id α α α 2 id Rcall that α and β can b thought of as th pmutations h v hv and h v hv and so Aut(G) Pm(G) = S n always holds. M. Macauly (Clmson) Chapt 8: Homomophisms Math 42, Summ I 24 5 / 5