# Groups and Symmetries

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1 Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group A group G, is a set G and an operation such that G is closed under and that: 1. There exists e such that e x=x=x e for all x G (existence of identity e ). 2. For every x G there exists x 1 such that x x 1 =e=x 1 x (existence of inverses). 3. For all x, y, z G, x y z = x y z (associativity). s D 3 is the group of symmetries of a regular 3-gon. D 3 ={e,r, r 2, s,sr, sr 2 }, D 3, with r =60 rotation clockwise and s=reflection about y-axis D n is the group of symmetries of a regular n -gon. The identity of G, is unique. Proof: Assume the e 1 and e 2 are identities of G,. e 1 =e 1 e 2 =e 2 e 1 =e 2. So there is only one identity. Given x G, x 1 is unique. Proof: Let y=x 1 and z=x 1 So x 1 is unique given x. but y z. Now y= y e= y x z = y x z=e z=z, so y=z. Contradiction! Definition: Commute If x y= y x then x and y are said to commute. Definition: Abelian If all elements in G, commute, then G, is said to be Abelian. Definition: Order (Group) The order of a group G, denoted, is the number of elements in G. s Q 8 =8, V 4 =4, D 3 =6, D 4 =8, Z =, Q =. Definition: Order (Element) The order of an element x G, written x, is the smallest positive integer n such that x n =e. s In D 3, r r r=r 3 =e so r =3. e =1. 1 of 17

2 In V 4 ={e,a,b,a b}, e =1, a a=e a =2, b b=e b =2, a b =2. In Q 8 ={1, 1,i, i, j, j, k, k }, 1 =1, 1 =2, i = j = k =4. Definition: Subgroup H is a subgroup of G iff H is a subset of G which is a group under the same operation as G. In D 4, {e, r, r 2, r 3 } is a subgroup since it is closed under ( r n r m =r n m ), has inverses ( r 1 =r 3, r 2 1 =r 2 ), and e {e, r,r 2, r 3 }. Definition: Proper Subgroup H is a proper subgroup of G iff H is a subgroup of G and H G, H {e}. Note Does G always have a proper subgroup? No. {e,r, r 2 } has no proper subgroups. Definition: Set of Generators S is a set of generators of a group G iff every g G can be expressed as: multiplication: g=s 1 m 1 s 2 m 2 s k m k addition: g=m 1 s 1 m 2 s 2 m k s k where m i Z and s i S (repetitions of s i 's are allowed). Any such combinations of s i 's is called a word. Definition: Relation A relation is an equation that tells use the rules for using in G,. If we specify that r 3 =e, s 2 =s, s r=r 2 s, the group generated by {r, s} is {e,r, r 2, s, r s, r 2 s}. We called this group D 3. So D 3 = r, s r 3 =s 2 =e,s r =r 2 s. Definition: Free Group A free group on n generators is a 1,,a n (a group with n generators and no relations). Definition: Cyclic A group (or subgroup) is cyclic iff it can be generated by only one element. a a 5 =e ={e, a,a 2, a 3, a 4 } is cyclic. Definition C n is the cyclic group of order n. C n = g g n =e. Definition: Infinite, Finite A group G is infinite iff =. A group G is finite iff =n. 2 of 17

3 Z, is infinite. In Z,, what does {2} generate? 0 2, 2 2, 2 2 2, 2 2, So 2 Z 2 ={2n n Z} is the set of even numbers. Theorem: The Subgroup Criterion If H G is a non-empty subset, then x, y H x y 1 H if and only if H is a subgroup. Proof: ( ) Assume x, y H x y 1 H. Let y=x, then x x 1 =e H. Let x=e, then y 1 H. So H is closed under inverses, i.e. y 1 H whenever y H. So take x and y 1 to be two arbitrary elements in H. Then x y 1 1 =x y H, so it is closed under multiplication. ( ) Assume H is a subgroup of G. So if x, y H, then y 1 H. Since H is closed under multiplication, x y 1 H. Definition Z n ={intergers mod n}. Let p be a prime. Z p has no proper subgroup. Proof: Let 0 n p. Since p is a prime, gcf n, p =1. So n 0mod p, 2n 0 mod p, etc., p 1 n 0 mod p. So Z p has no proper subgroup. HOMOMORPHISMS AND ISOMORPHISMS Definition: One-to-One 1:1 means f x = f y x= y, x, y. Definition: Onto Onto means if f : A B then for all b B there exists a A such that f a =b. Equivalently, f : A B is onto if and only if f A is all of B. Definition: Homomorphism f is an homomorphism iff f a b = f a f b where f : A, B,. Definition: Isomorphism f is an isomorphism iff f is a homomorphism, 1:1, and onto. Properties of Homomorphisms Let f : G, K, be a homomorphism. 1. Im f is a subgroup of K. 2. If H is a subgroup of G, then f H is a subgroup of K. 3 of 17

4 3. f sends inverses to inverses, i.e. f x = y f x 1 = y f sends identities to identities, i.e. f e G =e K. 5. If x 1 x 2 =x 2 x 1, then f x 1 x 2 = f x 2 x 1 = f x 1 f x 2 = f x 2 f x 1. Definition: Trivial Homomorphism f : G, {e} is called the trivial homomorphism. Properties of Isomorphism Let f : G, K, be a homomorphism. 1. All properties of homomorphism. 2. f preserves the order of elements, i.e. f x = y x G = y K. 3. = K. 4. G and K can be written be the same number of generators and the same relations. In other words, G and K have the same form. All groups are isomorphic to themselves. Definition: Symmetric Group The symmetric group (on n letters) S n is the group of permutations that permute up to n symbols/letters. s 23 4 :A B C D A D B C : A B C D E F G G A C B E F D :A B C D E B A E D C. Definition: Disjoint If there are no common numbers in two different sets of brackets, they are said to be disjoint. Definition: n-cycle A bracket with n distinct numbers is called an n -cycle. Note: A 2-cycle is also called a transposition. Is S 3 abelian? : A B C 123 C A B 23 C B A, but :A B C 2 3 A C B 12 3 B A C. So S 3 is not abelian. Conventions Write the smallest number first. So if 1 gets moved first, write 1, if 2 gets moved first, write 2, etc. Only write each number once to avoid confusion. Also end the bracket when the first repeating number appears. Using the convention gives a unique way of writing a permutation. However, there are many ways to write a permuation as a product of 2-cycles. 4 of 17

5 Theorem All n -cycles (except the identity e ) can be written as a product of 2-cycles and therefore all permutations. Theorem The 2-cycles generate S n. Question How big is S n? n!. Write as a product of 2-cycles = , = So = Definition: Even, Odd A permutation is even iff it can be written as the product of an even number of 2-cycles. A permutation is odd iff it can be written as the product of an odd number of 2-cycles. If can be written as an even number of 2-cycles, then it can never be written as an odd element. A n ={even permutations of S n } is a subgroup of S n, and hence a group. Note: A n = S n 2. Theorem The following are sets of generators for S n. All 2 cycles. 1 2, 1 3,, 1 n since a b = 1 a 1 b 1 a. 1 2, 2 3,, n 1 n since 1 k = k 1 k k 1 k. 1 2 and 1 2 n. Remarks If f : A B is an isomorphism, then g b is the preimage of f when f a =b and g b =a. g is also an isomorphism. Definition: Center The center of a group G is Z G ={z G z g=g z g G}, elements that commute with everything. Note: {e} is called the trivial center. Definition: Direct Sum The direct sum A B is the set of coordinates { a,b a A, b B}. Facts 5 of 17

6 1. Z G K =Z G Z K. 2. If H 1 is a subgroup of G and H 2 is a subgroup of K, then H 1 H 2 is a subgroup of G K. Remark Direct sums can be abelian or not, cyclic or not, etc., all the same definitions apply. Definition: Left Multiplication Function L g x =g x, g, x G is the left multiplication function. Lemma L g is 1:1 and onto when applied to the group G. Proof: Let L g x =L g y g x=g y g 1 g x=g 1 g y x= y, so L g is 1:1. Now assume L g is not onto. Then {L g x 1,, L g x n } is not the entire group. So L g x i =L g x j for some i j, then g x i =g x j x i =x j. Contradiction, so L g is onto. Theorem: Cayley's Theorem Let G be any group. There exists an isomorphism f such that f :G subgroup of S. In particular, if =n, G is isomorphic to a subgroup of S n. Proof: L g is 1:1 and onto, so L g permutes the elements of G. If =n, then L g S n. Let f :G S n, f g =L g. f a b =L a b =L a L b = f a f b, so f is a homomorphism. f is 1:1 since f x = f y L x =L y x g= y g, g G x e=y e x= y. Since f is 1:1 and takes n elements ( g G ) to n elements L g S n, f is onto. Hence f is an isomorphism. Theorem: Lagrange's Theorem Let H be a subgroup of G. Then H is a factor of. Proof: Let g 1 H but g 1 G, then g 1 H ={g 1 h h H } has the same number of elements as H (lemma applied to L g 1 ). Take g 2 H, g 1 H but g 2 G, then g 2 H = H. If is finite, then we get =k H where k is the number of g i 's, since G=H g 1 H g k H. Corollary Since g = g, then g is a factor of whenever g G. Let = p a prime number. If x G, then x =1 or x = p. If x =1, then x=e (i.e. x ={e} ). Otherwise G= x, which has no proper subgroup. This also shows that G is cyclic. NORMAL SUBGROUPS AND QUOTIENT GROUPS Definition: Partition A partition of G is a collection of disjoint subsets such that their union is G. Definition: Equivalence Relation 6 of 17

7 An equivalence relation ~ is a relation such that: 1. x~ x x G (reflexive). 2. x~ y y~x x, y G (commutative). 3. x~ y, y~z x~z x, y, z G (transitive). Definition: Conjugation If x=g 1 y g for some g G, then x is conjugate to y. Conjugacy is an equivalence relation. Proof: 1. x=e 1 x e. 2. Let x=g 1 y g. Let h=g 1. Then y=h 1 x h. 3. Let x=g 1 y g and y=h 1 z h. Then x=g 1 h 1 z h g=k 1 z k where k =h g. Definition: Equivalence Class An equivalence class is a complete set of elements that are equivalent to each other other. In other words, x is in the equivalence class of y if and only if x~ y. Equivalence relation partition all groups G in to equivalence classes. Proof: Assume two equivalence classes, say of x and y, are not disjoint. Then there exists z such that z~x and z~ y. Therefore x~ y and y~x, so they are the same classes. Contradiction. Hence two equivalence classes are disjoint. Now x~x, therefore any x G is in the equivalent class of x. So the union of all equivalent classes is G. Therefore equivalence classes partition G. Definition: Normal Subgroup H is a normal subgroup in G iff g 1 h g H for all g G and all h H. Definition: Normal Subgroup H is a normal subgroup in G iff g 1 H g {g 1 h g h H }=H for all g G. H ={e,r, r 2 } is a normal subgroup of D 3 Definition: Conjugacy Class The conjugacy class of x G is the set {g 1 x g g G} (equivalence class of x where the equivalence relation is conjugation). In D 3, the conjugacy class of r is {r, r 2 }. It is also the conjugacy class of r 2 by transitivity, since r ~r 2. The conjugacy class of e is {e} Definition: Normal Subgroup 7 of 17

8 H is a normal subgroup in G iff H is a union of conjugacy classes in G. H ={e, r,r 2 }={e} {r, r 2 } is a union of conjugacy classes, so H is a normal in G. Definition: Coset A coset of H in G is g H={g h h H } for some fix g G. s H ={s e,s r, s r 2 }={s, r 2 s,rs} is a coset of H. e H =H is also a coset of H. Definition: Index The index of H in G, denoted [G: H ], is the number of distinct cosets of H in G. Definition: Normal Subgroup H is a normal subgroup in G iff g H=H g for all g G. If G is abelian, H is always normal in G if H is a subgroup in G. Let H be a subgroup of G. The following are equivalent: 1. g 1 h g H for all g G and all h H. 2. g 1 H g=h for all g G. 3. H is a union of conjugacy classes in G. 4. g H=H g for all g G. 5. H is a normal subgroup in G, denoted H G. Proof: ( 2 1 ) Assume g 1 H g {g 1 h g h H }=H for all g G. Therefore elements of {g 1 h g h H } are also elements of H. ( 3 2 ) Assume H is a union of conjugacy classes in G, i.e. H = h H {g 1 h g g G}. So H = {g 1 h g g G, h H }=g 1 H g. ( 1 3 ) Assume g 1 h g H for all g G and all h H. So the conjugacy class of h is in H. Suppose H is not a union of conjugacy classes in G. Then there exists z H not in a conjugacy class. Now g 1 z g H, so the conjugacy class of z is in H. Contradiction. ( 2 4 ) g 1 H g=h g g 1 H g=g H H g=g H. Definition: Quotient Group The quotient group G/ H is the set of cosets {gh } g G with the operation x H y H =x y H where is the group operation from G. Note: This only works if H is a normal subgroup in G. Let H ={e,r,r 2 } D 3 be normal. Then D 3 / H ={eh, sh }. 8 of 17

9 Fact G /H = H (since L g is 1:1). Properties of Quotient Groups G/G={g G g G} {e}. If G is abelian, then G/ H is abelian. Proof: x H y H =x y H =y x H = y H x H since x y= y x. If G is cyclic, then G/ H is cyclic. Proof: G={g =1,, n}. The cosets in G/ H are {g H }, so g H generates G/ H whenever g generates G. If G is finitely generated, then G/ H is finitely generated. Proof: If g 1,, g n generate G, then g 1 H,, g n H generate G/ H. x H y H =x y H if and only if H is normal. Proof: ( ) x H y H =x y H x 1 x H y H =x 1 x y H H y H =y H, so h i y h j = y h k (fixing h i and h j we get h k ), so h i y=y h k h 1 j. Let h=h k h 1 j which can be anything in H. Then h i y= y h H y= y H, so H is normal. ( ) Clear from definition. Definition: Kernel, Image Let f :G G'. Then Ker f ={g G f g =e}, and Im f ={g ' G ' g G such that f g =g '}. Theorem: 1 st Isomorphism Theorem Let f :G G' be a homomorphism. Then Ker f G and G/Ker f Im f. Proof: Want: Ker f G. Let x Ker f. Then f g 1 x g = f g 1 f x f g = f g 1 e f g =e, so g 1 x g Ker f. Now, f z f e z = f e f z f e =e, so Ker f. Hence Ker f G. Want: G/Ker f Im f. Let K =Ker f. Let :G /Ker f Im f where x K = f x. Then is a homomorphism since a K b K = ab K = f a b = f a f b = a K b K. is onto Im f since is onto Im by definition of image, and Im =Im f by construction. is 1:1 since a K = b K f a = f b f a f b 1 =e f a f b 1 =e f ab 1 =e a b 1 K, so a b 1 K =K ab 1 K b K =K b K a K=b K. Hence :G /Ker f Im f is an isomorphism and so G/Ker f Im f. Corollary Let f :G G' be a homomorphism. Ker f ={e G } if and only if f is an isomorphism. Proof: ( ) Ker f ={e G }, then by the 1 st Isomorphism Theorem, G/{e G } Im f and so G Im f. Take : G /Ker f Im f defined by x K = f x as in the proof of 1 st Isomorphism Theorem. Here K={e }, so x = f x, and we already know that is an isomorphism, and hence f is an isomorphism. ( ) f is an isomorphism, so f is 1:1 and f e G =e G ', therefore Ker f can only contain one element, e G. Definition: Commutator Subgroup 9 of 17

10 The commutator subgroup [G,G ] is the subgroup of G that is generated by all x y x 1 y 1 for all x, y G. [G,G ] is abelian. Proof: Let H =[G,G]. x y x 1 y 1 H =H since x y x 1 y 1 H. So x y x 1 y 1 H y H =H y H x y x 1 H= y H x y x 1 H x H = y H x H x y H= y x H x H y H = y H x H. Hence H =[G,G] is abelian. If G/ H is abelian, then [G,G] H. Proof: x y H = y x H since G/ H abelian. So x y H x 1 H y 1 H =y x H x 1 H y 1 H x y x 1 y 1 H =H, so x y x 1 y 1 H. Therefore all generators of [G,G] is in H, so [G,G] H. Remark The commutator subgroup [G,G ] is the smallest subgroup H of G such that its quotient G/ H is abelian. Since bigger H implies few cosets, so G/[G,G] is the largest abelian quotient of G. Definition: Abelianisation The abelianisation of a group G is G/[G,G]. If A G, B G ', then A B G G' and G G ' A B G A G ' B. Proof: Define f : G G ' G A G ' B by f g,g ' = g A, g B. Then f g 1, g 1 ' g 2, g 2 ' = f g 1 g 2, g 1 ' g 2 ' = g 1 g 2 A, g 1 ' g 2 ' B = g 1 A, g 1 ' B g 2 A, g 2 ' B = f g 1, g 1 ' f g 2, g 2 ', so f is a homomorphism. Ker f ={ g, g ' G G' f g, g ' = e A,e B }, but f g, g ' = g A, g' B = e A, e B g A, g ' B, so Ker f ={ g, g ' G G ' g A, g ' B}= A B. Im f = { g A, g ' B G A G' B g, g ' G G' such that f g, g ' = g A, g ' B } =G A G' B, so f is onto. Definition: Lattice A group lattice is a diagram with subgroups as vertices and edges mean that the lower subgroup sits inside the upper subgroup. Theorem: 2 nd Isomorphism Theorem Let H, J be subgroups of G and J G. Then: 1. H J {h j h H, j J } is a subgroup. 2. H J G. 3. H J J H H J. Proof: e H e J =e G e G =e, so H J. Let g, g H J, g=h j, g= h j. Then g g 1 =h j j 1 h 1. Since j j 1 J, so 10 of 17

11 h j j 1 h 1 J because J G. Hence g g 1 J, so g g 1 =e j ' for e H and some j' J, and hence g g 1 H J. So H J is a subgroup by the subgroup criterion. Now, define f : H H J by f x =x e J. Then f x y =x e y e J =x e y e J J =x e J y e J = f x f y, so f is a J homomorphism. Ker f ={h H f h =e e J }={h H h J =e J }={h H h J }=H J. Im f = { h j J H J J h H such that f h =h j J } = { h j J H J J h H such that h e J =h j J } = H J J Therefore, by the 1 st Isomorphism Theorem, H J G and H J J H H J., so f is onto. Theorem: 3 rd Isomorphism Theorem Let H G, J G, and H J. Then: J 1. H G H. G /H 2. J / H G J. Note that H J since g H=H g, g G and H J, so j H =H j, j J G. Proof: Define f : G H G J by f x H =x J. Then f x H y H = f x y H =x y J =x y J J =x J y J = f x f y, so f is a homomorphism. Ker f = { g H G H f g H =e J } = { g H G H g J =e J } = { g H G H g J } = J H. Im f = { g J G J f g H =g J } = { g J G J g J =g J } =G J, hence f is onto. Therefore, by the 1st Isomorphism Theorem, J H G H and G /H J / H G J. Definition: Maximal Normal Subgroup H is a maximal normal subgroup of G if the only normal subgroups of G containing H are H and G. G has no proper normal subgroups if and only if H is a maximal normal subgroup in G. H Proof: ( ) Assume G H has no proper subgroups. Suppose H is not maximal. Let A G such that H A. Then A H since g H=H g, g G, especially for g A. So Contradiction. ( ) Assume H is a maximal normal subgroup in G. Suppose A H is a group. By part 1 of 3rd Isomorphism Theorem, G H has a proper normal subgroup A H A H G H. (can choose coset representatives such that A is a subgroup of G ). By part 2 of 3 rd Isomorphism Theorem, G /H A/ H G A. Since G A is isomorphic to a group, it is a group. Hence A G. But A H, so H is not maximal. Contradiction. GROUP ACTIONS Definition: Group Action 11 of 17

12 A group action G X X is such that g 1 g 2 x = g 1 g 2 x, where g 2, g 2 G ( G a group), x X ( X a set), is the group action, is the group operation. Definition: Orbit The orbit of x X is the set of images of x after being acted on by all g G. That is, O x ={g x g G}. Note: O x X. Definition: Stabilizer The stabilizer of x X is Stab G x ={g G g x=x}. Note: Stab G x G. Definition: Faithful A group action is called faithful if g x=x g=e. Remark A group action is faithful if and only if Stab G ={e}. Definition: Transitive A group action is transitive if for all x, y X there exists g G such that x=g y. Remark O X =X if and only if G X X is transitive. Definition: Centralizer In the case where G=G, X =G, g x=g 1 x g, the stabilizer of x X is Stab G x ={g G g 1 x g=x}. It is called a centralizer. The centralizer of A X =G is C G A ={g G g a=a g a A}. The centralizer of x X =G is C G x ={g G g x=x g}. Remark The centralizer of G is C G G =Z G, the center of G. Definition: Normalizer The normalizer of x G is N G x ={g G g 1 x g=x}=c G x. The normalizer of A G is N G A ={g G g 1 a g A}. Lemma Stab G x is a subgroup. Proof: e x= x, hence e Stab G x. Let g 1, g 2 Stab G x. Then g 2 x=x g 2 1 g 2 x=g 2 1 x e x=g 2 1 x x=g 2 1 x, so g 2 1 Stab G x. So g 1 g 2 1 x=g 1 g 2 1 x =g 1 x=x, hence g 1 g 2 1 Stab G x. So by the subgroup criterion, Stab G x is a subgroup. 12 of 17

13 Theorem: Orbit-Stabilizer Theorem O x = Stab G x. Proof: Fix x X. O x ={g x g G}. Define f :O X Stab G x by f g x =g H x where H x =Stab G x. f is onto since all g H x will have preimage g x. Now let g 1 H x =g 2 H x. Then g 1 =g 2 h for some h H x, so = g 1 x=g 2 h x =g 2 x. Hence f is 1:1. So if, then O x = G H x G H x. Theorem: Class Equation Let G=G, X =G, g x=g 1 x g. Then other. = Z G x Stab G x Stab G x with no repetition of x 's conjugate to each Proof: Conjugacy class of z Z G is {z}. Conjugacy class partition G, so = Z G conjugacy class of x x Z G. Now, O x ={g x g G}={g 1 x g g G} is the conjugacy class of x, and O x = by the orbitstabilizer theorem. Hence = Z G Stab G x x Stab G x Stab G x. Lemma O x is a factor of. Proof: O x = Stab G x by orbit-stabilizer theorem. Stab G x is a subgroup of G, and so Stab G x is a factor of by Lagrange's theorem. If p is prime, then = p k if and only if Z G 1. Proof: Let G=G, X =G, g x=g 1 x g. The by the class equation, = Z G O x, so O x has p as a factor. Suppose Z G =1. Then p k=1 i Z G = p for some k. p i, but p should be a factor of both sides. Hence Z G 1. In fact, Theorem: Cauchy's Theorem Let =k p m where p is prime and p is not a factor of k. Then there exists x G such that x = p. Proof: Let X ={ x 1,, x p x 1 x p =e, x i G } the set of p -tuples in G such that x 1 x p =e. X is a multiple of p. Let m Z p act on X by sending m x 1,, x p to x m 1,, x p, x 1,, x m. Each order of an orbit has either 1 or p p -tuples. Now e,, e has orbit { e,, e }. Since orbits partition X, there exists some other orbit hat has one element as well. Call the element in this orbit g 1,, g p. m g 1,, g p = g 1,, g p m Z p, so g 1 = =g p and g 1 g p = g 1 p =e. Therefore x=g 1 is the desired element. Remark 13 of 17

14 Recall that N G A ={g G g 1 a g A} and C G A ={g G g a=a g a A} for a subgroup A G. In general, if A G then N G A =G. THE SYLOW THEOREMS Theorem: Sylow Theorem 1 Let =k p m where p is prime and p k. Then G contains a subgroup of order p m. Proof: Let X be the collection of subsets of G of p m elements. Use induction on. Induction hypothesis: G has a subgroup of order p m. If =1, then G={e }, therefore vacuously true. If =2, then G={e,a}, therefore G is its own subgroup of order 2 1. Now let =n=k p m. Assume induction hypothesis for all k p m. Case 1: If p is a factor of Z G, then Cauchy's Theorem implies Z G has an element of order p (since Z G is a group); call it g. g generates a subgroup of order p ; call it N. Since N Z G, x n=n x, n N, x G, hence x 1 n x N, n N, x G, so N G. Now G N = N = k pm p =k pm 1, so by induction hypothesis G N has a subgroup, say P N, of order pm 1. P = P N N = P N N = pm 1 p= p m. Therefore P is a subgroup of G of order p m. Case 2: If p is not a factor of Z G, then let g 1,,g r be representatives of distinct conjugacy classes of G Z G. By the class equation, = Z G. p cannot be factor of all or else p i C G g i C G g i is a factor of Z G. Therefore there exists a g i such that C G g i = k pm l p = k where gcf l, p =1. By the m l Orbit-Stabilizer Theorem, C G g i = conjugacy class of g i. Let H =C G g i. Then H =l p m k p m since g i Z G, so H. By induction hypothesis, H has a subgroup of order p m, which is also a subgroup of G. Definition: Sylow p-subgroup Let =k p m where p is prime and p k. A subgroup of G of order p m is called a Sylow p -subgroup. Theorem: Sylow Theorem 2 Let =k p m where p is prime and p k. Any two Sylow p -subgroups are conjugate, i.e. there exists g G such that g 1 P g=q for all P and Q subgroups of order p m. Theorem: Sylow Theorem 3 Let =k p m where p is prime and p k. If n p is the number of Sylow p -subgroups, then n p 1mod p and n p k. Proof: Let H 1,, H t be the subgroups of G of order p m. Now let H 1 act on {H 1,, H t } by conjugation (i.e. h H j =h 1 H j h ). Then Stab H 1 H j =K j =H 1 H j (**). Now, K 1 =H 1, so O H 1 =H 1. Now if j 1, K j is a smaller power of p than p m. Then the Orbit-Stabilizer Theorem implies O H j is multiples of p since they are factors of H 1. Since t is the number of subgroups being acted on, so t=1 multiples of p 1 mod p. Hence the number of Sylow p -subgroups is n p 1mod p. Now let G act on {H 1,, H t } by conjugation. Suppose H r (for some r ) is not in the orbit of H 1. Now let H r act on {H 1,, H t } by conjugation. The G -orbit of H 1 is partitioned into H r -orbits, the size of which are multiples of p by 14 of 17

15 the Orbit-Stabilizer Theorem since H r is not in H 1 's orbit. So the G -orbit of H 1 has 0 mod p elements in it, which contradicts n p 1mod p. Hence H r cannot exists and the G -orbit of H 1 is {H 1,, H t }. In other words, O H 1 ={g H 1 g G}={g 1 H 1 g g G}={H 1,, H t }. So given any H i and H j, H 1 =g 1 i H i g i and H 1 =g 1 j H j g j, so g 1 i H i g i =g 1 j H j g j H i = g i g 1 j H j g j g 1 i =g 1 H j g for g=g j g 1 i. Therefore Any two Sylow p -subgroups are conjugate. Now, Orbit-Stabilizer Theorem implies O H 1 = Stab G H 1 and so t=n p is a factor of =k p m. Since we know t 1mod p, t is not a factor of p m, hence t is a factor of k. Therefore n p k. (**) K j =Stab H 1 H j ={h H 1 h 1 H j h=h j }, so K j H 1. H 1 H j K j, hence K j H j =H j K j and so K j H j is a subgroup. Also, H j K j H j since H j k =k H j. So by the 2 nd Isomorphism Theorem, K j H j K j. So H j K j H j K j H j = K j H j K j H j = pr, but H j K j H j and H j = p m, so K j H j =H j. We have K H 1 H j K j, so K j =H 1 H j. Lemma If H, K are subgroup of G, and H K=K H, then H K is a subgroup. Proof: Let h 1 k 1, h 2 k 2 H K. Then h 1 k 1 k 2 1 h 2 1 =h 1 k 3 h 2 1 =h 1 h 3 k 4 =h 4 k 4 H K. Also, e H K. So the the Subgroup Criterion, H K is a subgroup. Corollary (of Sylow Theorem 2) If n p =1, i.e. H is the only Sylow p -subgroup of G, then H G. THE FUNDAMENTAL THEOREM OF FINITELY GENERATED ABELIAN GROUPS Theorem: The Fundamental Theorem of Finitely Generated Abelian Groups Let G be abelian and finitely generated. Then G Z m1 Z mt Z s where m i m i 1, m 1,, m t, s Z, s m 1,, m t, s 0, Z s =Z Z. Here m 1,,m t are called torsion coefficients, and s is the rank. Theorem Z n Z m =Z nm if and only if gcf n, m =1. Definition: Canonical Form We say Z m1 Z mt Z s is a group written in canonical form iff m i m i 1. Question How to to between canonical form and generators-and-relations form? Given canonical form Z m1 Z mt Z s, let x i= 0,, 0, 1, 0,, 0, 0,,0 for i=1,, t, and t j 1 s j i 1 t i s y j = 0,,0,0,,0,1,0,,0 m for j=1,, s. Then x 1,, x t, y 1,, y s abelian, x i=e i for i=1,, t is the generators-and-relations form. Given generators-and-relations form, use the following steps. 15 of 17

16 1. Rewrite the relations in additive notation, ignoring the ones specifying abelianism. 2. Write out the coefficient matrix, where entries are coefficients of each generator. 3. Diagonalize using the three rules: switch any two rows or columns, multiply -1 to any row or column, add an integer multiple of any row to row, or column to column. 4. Each non-zero entry on the diagonal is a torsion coefficient. A zero means a copy of Z, and a 1 means one too many generator. AUTOMORPHISMS Definition: Automorphism An automorphism is an isomorphism from a group G to itself. Definition: Automorphism Group Aut G is the set of all automorphisms of G. For (i.e. finite groups), Aut G is a group under function composition. Proof: Let f 1,, f n Aut G. f 1 x =x is the identity since f i f 1 x = f i f 1 x = f i x and f 1 f i x = f 1 f i x = f i x. Let f i Aut G. f i 1 exists and is 1:1 onto since f i is 1:1 onto on a finite set to itself. We know f i a b = f i a f i b since f i a homomorphism, so f i 1 f i a f i b = f i 1 f i a b =a b= f i 1 f i a f i 1 f i b. Hence f i 1 is a homomorphism, 1:1 and onto, so f i 1 Aut G. f i f j f k x = f i f j f k x = f i f j f k x, so associativity holds. Let f i, f j Aut G. Then f i f j is 1:1 since f i and f j are, and f i f j is onto since f i and f j are. f i f j ab = f i f j a b = f i f j a f j b = f i f j a f i f j b = f i f j a f i f j b, so f i f j is a homomorphism. Hence Aut G is closed under composition. Therefore, Aut G is a group. Definition: Characteristic H is characteristic (as a subgroup of G ), denoted H char G, iff f H = f, f Aut G. Theorem 1. Characteristic subgroups are normal. 2. If H is the only subgroup of a given order, then H char G. 3. If K char H and H G, then K G. Definition: Inner Automorphism Inn G is the set of inner automorphisms of G. An inner automorphism is a function f such that f x =g 1 x g for a fixed g. f g Inn G is an isomorphism. 16 of 17

17 Proof: f g is 1:1 since L g and R g are. f g is onto since only use g 's that give onto f g 's. f g is a homomorphism since f g a b =g 1 ab g=g 1 a g g 1 b g= f g a f g b. 17 of 17

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