Lat time: Let G ü A. ( ) The obit of an element a P A i O a tg a g P Gu. ( ) The tabilize of an element a P A i G a tg P G g a au, and i a ubgoup of G. ( ) The kenel of the action i ke tg P G g a a fo all a P Au apa G a, and i a nomal ubgoup of G (ince ke kep'q). ( ) An action i faithful if and only if the kenel i tivial. ( ) An action i tivial if g a a fo all a P A, g P G. ( ) The action of G on A i called tanitive if thee i only one obit. Theoem (Obit/tabilize theoem) Fo each a P A, leto a be the obit in A containing a. Thenthee i a natual bijection O a ØtgG a g P Gu, othat O a G : G a. Lat time: S n xt y whee T tpijq i j nu. Claim Fix P S n.ifwehave ` fo,...,` P T, and t t t m fo t,...,t m P T, then ` and m ae eithe both even o both odd. (i.e. Evey expeion of a a poduct of tanpoition ha length of the ame paity.) Poof by way of the action on polynomial: Recall that S n act on Zx,...,x n by ppx,...,x n q ppx pq,...,x pnq q. Let π i j n px i x j q px loooooooooooooooooooooomoooooooooooooooooooooon x qpx x q px n x n q npn q{ tem
Conide π px piq x pjq q. i j n Since P S n i a bijection on n, it i alo a bijection on t ize- ubet of n u.soeveytempx i x j q of will appea in,eithea px i x j q o px j x i q px i x j q. Fo example, if n and pq, then px x q px x q px x q px x q px x q px x q :p q px x q px x q px x q px x q px x q px x q So. Specifically, we have p q #t i j n pjq piqu. Let p q # ` if, if. o that p q.wecall p q the ign of. Popoition The map : S n Ñt uˆ i a homomophim. Popoition All tanpoition have negative ign, i.e. ppi jqq. Theefoe, () the homomophim i ujective fo all n, and () fo any expeion of P S n a the poduct of tanpoition, the paity of the length of that poduct i detemined; namely the length i even if p q, and odd if p q.
Definition We call a pemutation even if p q and odd if p q. The altenating goup A n i the kenel of : S n Ñt uˆ, i.e. A n t P S n i even u. Caution: even/odd A pemutation i even if it i a poduct of an even numbe of tanpoition, and i odd if it i a poduct of an odd numbe of tanpoition. So an m-cycle i even if m i odd, and i odd if m i even! Example A tu A tu A tuyt-cycleu t, pq, pqu A tuyt-cycleuyttwo dijoint -cycle u t, pq, pq, pq, pq, pq, pq, pq, pq, pqpq, pqpq, pqpqu
Fact about A n and ign ( ) A n ú S n (ince A n kep q) ( ) S n : A n (S n {A n Z by t io thm) ( ) S n {A n i imple ( ) S n A n todd pemutation in S n u pqa n (ince coet of A n,ofwhichtheeae,patitions n,andpq RA n ). ( ) The ign of pemutation multiply like the paity of intege add: pevenqpevenq pevenq poddqpoddq pevenq pevenqpoddq poddq poddqpevenq poddq (ince i a homomophim) ( ) A cycle i even if and only if it i of odd length. So a pemutation i even if and only if it cycle decompoition ha an even numbe of even-length cycle. ( ) Fo n, A n i non-abelian. (A A and A Z.Butifn, thenpq, pqpq PA n.) ( ) Fo n, A n i imple. (Poven in.)
Exitence and uniquene of cycle decompoition We ve been uing the fact that fo all P S n, Exitence: can be witten a the poduct of dijoint cycle (called it cycle decompoition), and Uniquene: it decompoition i unique up to pemutation of cycle and cyclic pemutation within cycle. Poof: Let n t,...,nu and let S n act on n in the natual way. Fix P S n, and conide the obit of G x y on n. Fo example, if n and pq, then xpqy t, pq, pqu, and the obit ae O t,, u, O tu, O tu. Fix an x Pn. Recall the bijection t left coet of G x u Ø O x i G x Ø i x Exitence and uniquene of cycle decompoition In come the good tu about ubgoup and quotient of cyclic goup... Fo each x Pn, inceg x G x y, ( ) G x i cyclic and nomal. ( ) Theode gg x in G{G x i the mallet poitive d uch that d P G x, and the ditinct coet of G x ae G x, G x,..., d G x. So the element of O x ae x, pxq,..., d pxq. In othe wod, act a the d-cycle px pxq d pxqq on O x. Moeove, the obit in n patition n. So, fo each obit O x, thi agument poduce a O x -cycle; and i equal to the poduct of thoe (dijoint cycle). Thi give u exitence.
Example of exitence poof Let, o that,, and, and o G x y t,,, u. Then the obit ae O t,,, u, O t, u, and tu. Example of exitence poof G x y t,,, u O t,,, u, O t, u, and O tu. Fix x. We have G tu, whichinomaling x y. The ode of G in G{G i the mallet poitive powe d uch that d P G, od. The ditinct coet of G in G ae G, G, and the element of O ae, pq, G, and pq, and G, pq. The coeponding cycle in (the one that contain ) i p q.
Example of exitence poof G x y t,,, u O t,,, u, O t, u, and O tu. Fix x. We have G t, u x y,whichinomalin G x y. The ode of G in G{G i the mallet poitive powe d uch that d P G,od. The ditinct coet of G in G ae and the element of O ae G and G, and pq. The coeponding cycle in (the one that contain ) i p q. Example of exitence poof G x y t,,, u O t,,, u, O t, u, and O tu. Fix x. We have G t,,, u x y, whichinomalin G x y. The ode of G in G{G i the mallet poitive powe d uch that d P G,od. So the only coet of G in G i G, and the only element of O i. So the coeponding cycle in (the one that contain ) i pq. Putting it all togethe, we have pqp qpq.
Exitence and uniquene of cycle decompoition Fo uniquene, note that detemined the cycle. detemined the obit, which The feedom:. The ode that you lit the obit (eaanging cycle).. Choice of epeentative of O: If you chooe i pxq intead of x, you get element i pxq, i` pxq,..., d pxq,x,..., i pxq, which i the ame a cycling the entie in a cycle. Goup act on themelve by left multiplication A goup G act on itelf by left multiplication: g a ga fo all g, a P G. Fo example, the action of D 8 on itelf by left multiplication look like:
Goup act on themelve by left multiplication A goup G act on itelf by left multiplication: g a ga fo all g, a P G. Lemma Let G be a goup, and let G act on itelf by left multiplication. Then. G act tanitively on itelf,. the tabilize of any a P G i, and. the kenel of the action i (the action i faithful). The induced map ' : G Ñ S G i called the left egula epeentation. Coollay (Cayley theoem) Evey goup i iomophic to a ubgoup of a (poibly infinite) ymmetic goup. In paticula, G i iomophic to a ubgoup of S G S G. In othe wod, evey goup i iomophic to a pemutation goup.
A bit moe ophitication... Let H G and conide the action of G on the left coet of H induced by left multiplication: g pahq pgaqh fo all g, a P G. Theoem Let G be a goup with H G, andletg act on A tah a P Gu by left multiplication. Then. G act tanitively on A,. the tabilize of H i G H, and. the kenel of the action i apg aha, which happen to be the laget nomal ubgoup of G contained in H. Coollay If G i finite of ode n, andp i the mallet pime dividing n, then any ubgoup of index p i nomal.