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Algebra II

by: Miss Hillary Grady

Algebra II MATH 430

Miss Hillary Grady

GPA 3.96

Christopher Vinroot

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Christopher Vinroot
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This 6 page Class Notes was uploaded by Miss Hillary Grady on Thursday October 29, 2015. The Class Notes belongs to MATH 430 at College of William and Mary taught by Christopher Vinroot in Fall. Since its upload, it has received 24 views. For similar materials see /class/231140/math-430-college-of-william-and-mary in Mathematics (M) at College of William and Mary.

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Date Created: 10/29/15
Group Actions Math 430 Spring 2009 The notion of a group acting on a set is one which links abstract algebra to nearly every branch of mathematics Group actions appear in geometry linear algebra and differential equations to name a few Group actions are a fundamental tool in pure group theory as well and one of our main applications will be the Sylow Theorems Sections 36 and 37 in P raleigh These notes should be used as a supplement to Section 16 of P raleigh7s book Some of the notation here will differ from the notation in Fraleigh but we will attempt to point out whenever this happens Let G be a group and let X be a set Let SymX denote the group of all permutations of the elements of X written as SX in Fraleigh So if X is a nite set and le n then SymX Sn We will give two equivalent de nitions of G acting on X De nition 1 We say that G acts on X if there is a homomorphism b G a SymX One way of thinking of G acting on X is that elements of the group G may be applied to77 elements of X to give a new element of X The next de nition takes this point of view De nition 2 We say that G acts on X if there is a map gtk G gtlt X a X so that ifg E G and z E X then gx g x E X such that i For every gh E G x E X we have gh x ggtk hgtkx ii For every x E X e gtk z x where e E G is the identity If the group G acts on the set X we will call X a G set Note that Fraleigh often writes 9x for g gtk s where g E G and z E X Before giving examples we need to show that the two above de nitions actually de ne the same notion Theorem 1 De nition 1 and De nition 2 are equivalent Proof First assume that G and X satisfy De nition 1 so that we have a homomorphism b G a SymX We now show that G and X must also then satisfy De nition 2 We de ne a map gtk G gtltX a X by ggtkz g First for every 9 h E G x E X using the fact that b is a homomorphism we have 911 96 MgM96 459 0 WWW 9 h9 9 h 96 so that gtk satis es condition of De nition 2 Also since b is a homomor phism e is the triVial permutation where e E G is the identity element So e gtk z ez x which is condition ii of De nition 2 Thus G and X satisfy De nition 2 Now suppose G and X satisfy De nition 2 so that we have a map gtk G gtlt X a X which satis es and ii We de ne a map b G a SymX by g g gtk x We rst show that this is well de ned that is g is actually a one to one and onto map from X to itself To show that g is onto let x E X and consider 9 1 gtk z E X Then we have gg 1 gtk x g gtk 9 1 gtk x 9971 gtk z e gtk z n so g is onto To show that g is one to one suppose that we have gx gy for my 6 X so that g gtk z g gtk y Using both conditions i and ii of De nition 2 we have 94949 9 1gy9 1996 9 19y 2 em My 2 z 1 Finally we show that b is a homomorphism Let g h E G x E X We have 9h96 9h 96 9 h 96 9 h9 W9 0 WWW Thus G and X satisfy De nition 1 D Now that we have a few ways of thinking about group actions lets see some examples Example 1 As mentioned before we may take X 12n G Sn SymX and b Sn a Sn to be the identity map Example 2 Let X R and G GLnR and for A E G o E X de ne A gtk o A1 That is we let G act on X as linear transformations Example 3 Let X be a unit cube sitting in R3 and let G be the group of symmetries of X which acts on X again as linear transformations on R3 Example 4 Let X be a group H and let G also be the same group H where H acts on itself by left multiplication That is for h E X H and g E G H de ne ggtkh gh This action was used to show that every group is isomorphic to a group of permutations Cayley7s Theorem in Section 8 of Fraleigh Before de ning more terms Well rst see a nice application to nite group theory Theorem 2 Let G be a hite group arid let H be a subgroup ofG such that G H p where p is the smallest prime dividing Therz H is a normal subgroup of G Proof We let X be the set of left cosets of H in G From the proof of Lagrange7s Theorem Section 10 of Fraleigh we have le G H p and so SymX g Sp We de ne an action of G on X by g gtk aH gaH for g E G and aH E X That is we let G act on the left cosets of H in G by left multiplication This satis es De nition 2 since for any g1g2a E G we have 9192 gtk aH glggaH and e gtk aH aH From Theorem 1 and since SymX 2 SP we have a homomorphism b G a Sp For any g E Gg Z H we have ggtkH gH 31 H and so bg cannot be the triVial permutation of left cosets of H in G that is g Z ker when g Z H We must therefore have kerb S H From the rst isomorphism theorem for groups we have Gkerb g imb where imb G is a subgroup of Sp So we have M lker l lGker l lSpl pl Note that p is the largest prime dividing pl and p2 does not divide pl while p is the smallest prime dividing Since ker S H and H is a proper subgroup of G we cannot have G kerb that is G ker 74 1 The only possibility is that lGker l G kerb p since this is the only divisor of lGl which divides pl We now have 1G1 lker l so that lker l Since kerb Q H we must have H kerb which is a normal subgroup of G 10GHE 1G ker l H We now de ne a few important terms relevant to group actions De nition 3 Let G be a group which acts on the set X For x E X the stabilizer ofz in G written stabgx is the set of elements 9 E G such that g gtk z x In symbols stabg g E G l ggtk z ln Fraleigh this is called the isotropy subgroup of s and is written Gm we show below that this is actually a subgroup of G For m E X the orbit ofz under G written orbgz is the set of all elements in X of the form 9 gtk z for g E G In symbols orbGz g gtk x l g E G Fraleigh uses the notation Gs for the orbit of z under G Example 5 Let G 1 1 2 3 4 6 3 6 4 1 23 4 6 1 23 6 4 and let b G a 6 a 04 be the natural injection as G is a subgroup of 56 Then G acts on 12345 6 First note that since 5 is xed by every element of G we have stabg5 G and orbg5 We also have stabg3 stabg4 stabg6 1 2 stabg1 stabg2 3 4 6 orbg3 orbg4 orbg6 346 orbg1 orbg2 1 2 Example 6 Let G be any group and we let G act on itself by conjugation That is for ga E G we de ne 91 gag l We rst check that this satis es 4 De nition 2 First we have e gtk a eae l a Now let gha E G Then we have gh gtk a ghagh 1 ghahilgil g gtk h gtk a so this is indeed a group action If we x an a E G we see that the orbit of a is 0rbaa 90971 l 9 E G which is called the conjugacy class ofa in G If we look at the stabilizer of a in G we have stabaa 9 E G l 9094 a which is the centralizeiquot ofa in G also written Gaa The next Lemma shows us that stabilizers of group actions are always subgroups and so in particular centralizers of elements of groups are subgroups Lemma 1 IfG acts on X and z E X then stabgz is a subgroup of G Proof Let x E X Since e gtk z x we know that e E stabgz and so the stabilizer of z in G is nonempty Now suppose gh E stabaz Since ggtkzz we have g 1gtkggtkz g 1z g 1gz g 1gtkz egtkx g 1gtkz g 1z z So g 1 E stabaz We also have ghgtkzggtkhgtkzggtkzz so gh E stabaz Thus stabgz S G D The next result is the most important basic result in the theory of group actions Theorem 3 OrbitStabilizer Lemma Suppose G is a group which acts on X F07quot any x E X we have lorbgzl G stabGz which means that the cardinalities are equal even when these are in nite sets IfG is a nite group then lGl lstabazl lorbgzl Proof Fix z E X From Lemma 1 stabg is a subgroup of G and we recall thatG H denotes the cardinality of the set of left cosets of H in G Let IC denote the set of left cosets of H in G De ne a function f orbaz a IC by fggtm gH First we check that f is well de ned and at the same time check that f is injective lf 9192 6 G 91 a 92 a E orbax if and only if gz lgl a a iff 92491 E stabgx H which is equivalent to 92H 91H So 91 gtk z 92 gtk x if and only if fgL gtk a f2 gtk z and f is well de ned and injective Also f is onto since for any 9H 6 IC fg gtk a gH Thus f gives a one to one correspondence and so lorbgxl liq G stabGx When G is nite it follows from the proof of Lagrange7s Theorem that G stabaz lGllstabgl So in this case lGl lstabaw lorbgw D Next we connect the concept of a group action with the important notion of an equivalence relation Theorem 4 Let G be a group which acts on a set X and for my 6 X de ne z N y to mean that there is a g E G such that g gtk z y Then N is an equivalence relation on X and the equivalence class ofz E X is orbg Proof We must check that N satis es the re exive symmetric and transitive properties First for any x E X we have e gtk z a where e is the identity in G and so z N z and the re ective property holds Next if z N y then there is a g E G such that g gtk z y It follows from De nition 2 that we then have 9 1 gtk y a so that y N z and the symmetric property holds Now assume z N y and y N z where g gtk z y and h gtk y 2 Then from De nition 2 h gtk g gtk a hg gtk z z and so z N z and transitivity holds So N is an equivalence relation From the de nition of an equivalence class if z E X then the class of z istheset yEX l zwyy X l yggtkxforsomeg G This is exactly the de nition of the orbit of z under G D We conclude with one more application to group theory this time to the


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