gpc x All 4i s at gpcx 0CxsfcyjVx.yE Imf 0cg IgE G C H is a Facts HMIEL cxgj it 7 X CG 4 gpc x G Escala c 23 G cyclic G E CZ t G Zaz t

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Gwen Brown
5 years ago
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1 MidtermRevieir A group is a set G equipped with a G x G G such that x y say c notation binary operation ht x y z e G associative Y Ky 2 x Cyz 7 Be E G s t see ex x tf x c G identity 3 Given see G ye G s t xy 9 written Example CZ 1 also X Gluck 74nz.tl Facts y he identity and inverses are It ay yx si yx e unique V x.ge G we say G AE inverses H C G a subgroup Y ee H I x.se a III II.am E H si e H left coset d H containing x Given ace G a H och h E H C G Facts Y x H l x c G Form a partition of yh x H I y z I HIC IHI Ix Hl Axe H c H G

2 i La em Glo HMIEL et X C G just a subset Define X Sai Ixex gpc x All 4i s at mp elements at Xu X use subgroupgenerated by Facts gpcx C G is a subgroup We G is say finitelygenerated Ct g it 7 X CG Y IX s and 4 gpc x G Escala c 23 r All G cyclic I see G s't f a gp 5 t G cyclic G E CZ t G Zaz t Given x C G Ord a Igp a I min in c IN s t am e I a I and cyclic I see G st ord x I G Facts t G cyclic and ml 1Gt F H C G a subgroup s E IH l m Abelian y Every subgroup of cyclic group is cyclic Let G and H be two groups A homomorphism From G to H is a map G H s.to cxgj P 0CxsfcyjVx.yE in G in H G H a homomorphism 10Cee ex and fix I cx5 V xeg Isomorphism Bijective homomorphism G EH F G H an isomorphism say G isomorphic to H Imf 0cg IgE G C H is a subgroup injective G E Ion lo

3 binaryoperation Let 5 be a set d functions Permutation group at 5 s t S s I I bijection Symu Z 1,2 u Anne N th finite symmetric group An action of G ou s is a s t map Y E Cs s and 4 GUKSI glues Vh.geGcses Equivalently G s s g s f g s an action of G ou S is a homomorphism 4 G ECS Say action is faith if 9 injective I Example Left G XG G regular representation Cg x ga t Cayley's heorem G is isomorphic to a in a subgroup of ECG I G l u G isomorphic to a subgroup of Symu Let G act on 5 orbit of S Given SE S orbis gcs I got G C 5 Feat We he orbits 7am a partition A ie t e orbis orb t Wbls the action is transitive orb s S fses say given s t E S 7g E G s t gcs C stab g e G I g s s C G A stabilizer subgroup of s bit Stabiliser 1Gt c If I Istabcs lovbcssl louses 1Gt and I stab G A D S Lagrange S s

4 Isymul ul cydent length r 5 Cycle notation a.az ar art as a n e Fact Every e Symu can be written as a product of disjoint cycles he lengths at the cycles give a partition at n called the cycle structure of 0 b 123 E.g 4S C Syms has cycle strueta 3,2 32 l S 4 C Syms has cycle structure 41,1 I nd o LCM at cycle lengths in disjoint de Lowest common multiple ab transposition Fat c Symon can be written as a product of transpositions he number transpositions must always either be odd or even We call 0 odd or every respectively Attu EE Sigma even C Symn alternating subgroup generated by Alta ISymul ul length 3 abc Z Z cycles at Isom R IR R l DCI y n H I I C R group at isometries et IR d doe cy

5 ur XC 112 just a subset Sym CX EE Isom IR l Symmetry grop at X n 0 permutes X Du Sym Regular n gou E g D Sym I nth Dihedral Group Rotcx7 snfy.ee m Lett asset c Rot X at reflections Du e E E E I to to to rotation by2in Reflection clockwise ord n nd 2 b d e d f de 2 Du non Abelian and I Dul 2h written No G I Let N C G be a subgroup We say N natural in G it ne N GE G gag e N Facts N a gn Ng jehtamt We G is say simple 17 No G N Ee or G Noe Fades GIN x GIN GIN XN y N cry N is well denied and gives 97N the structure a We group call it the quotient group he map G is called the quotient se homomorphism an

6 3rd Isomorphism heorem NO G here is a bijection subgroups at a containing µ H Y HINO G IN H G and NYC subgroups at Gen HIN µ FH Let G H be a homomorphism kerf GEE I 10cg e 3 C G kernel at 0 Fact kerf o G 1st Isomorphism heorem G Keef In 0 x kerf dex is a well delivered isomorphism If I G I c the IGI Ikerd l Italo Directproducts Let H Hu be groups Composition in It x Hr x Hr has a group structure given by it I hi hz h r g ga s gr hag shag h u g r H Hz x Hr Direct Product at A Hn H I e z e H hi EH i e v f hi C H C Hix x Hr Subgroup at A Hzx x Hu

7 Direct sums subgroups It Hu C G G It HzD It Y Given g E G I l hi c Hi such that g h h has hr 7 hi hj hj hi t hi C Hi hj c Hj i j Fact iuiteyg enerat.cl H Hr Hix Hz x x H ehiangroups G F g Abelian group 2G see G l and Lx c c C ension subgroup 7 g G and torsion free Etta is f Ite g Free Abelian Gita has a 2 basis rank G rank size of a GGG 2 basis Feat F F C G a t g ree Abelian group s t rank F rank G u and G F to th I 2 x tf te GIF tf 7 g and tension It G Ie Let It G I Pi PI Pi distinct i E IN primes the p th p where E pi a e th l p x o c th

8 G pi C Cd where i are cyclic subgroups he Ci are unique up to isomorphism Ici l power G p Zip ftp dz at pi Putting all this together gives Structure heorem For Finitely Generated Abelian Groups Let G be a Finitelygenerated Abelian group hen F Ci Cz Cu cyclic groups such tht G I C X x x Cn z Ci is either Z t n Zlpke 1 Fu p prime and K E IN 3 If G D X x Dan is another such expression Zhen n m and after reading C i D

Groups and Symmetries

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