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Let n € S8 be the permutation defined by (1) = 3, (2) = 8, (3) = 5, (4) = 4, TT(5) = 6, (6) = 1, \T(7) = 7, (8) = 2. a. Prove that T has order 4. b. Let G be the subgroup fo S8 generated by TT, i.e., G = (IT) = {e, 11, T2,,T3}. Describe all of the orbits of G, as was done in Example 6.11. c. Let X = {1, 2, 3, 4, 5, 6, 7, ,8}, so G acts on X. For each k € X, describe the stabilizer Gk. d. Use your data from (b) and (c) to explicitly verify the orbit-stabiliser counting formula Exercises Section 6.1. Normal Subgroups and Quotient Groups 6.1. Let : G ! G0 be a homomorp hhiissmm (a) Prove that the image (G) = (g) : g 2 G }is a subgroup of G0. (b) Suppose that G is a finite group. Prove that #G = # (G0)  #ker( ): 6.2. Let G be a group, let H  G and K  G be subgroups, and assume that K is a normal subgroup of G. (a) Prove that HK = fhk : h 2 H; k 2 Kg is a subgroup of G. (b) Prove that H \K is a normal subgroup of H, and that K is a normal subgroup of HK. (c) Prove that HK=K is isomorphic to H=(H \ K). (Hint. What is the kernel of the surjective homoorphism H ! HK=K?) 6.3. Let G be a group, let K  H  G be subgroups, and assume that K is a normal subgroup of G. (a) Prove that H=K is naturally a subgroup of G=K. (b) Conversely, prove that every subgroup of G=K looks like H=K for some subgroup H satisfying K  H  G. Section 6.2. Groups Acting On Sets and a Counting Theorem 6.4. Let G be a group, let X be a set on which G acts, and let x 2 X. Prove that the stabilizer Gx of x is a subgroup of G. 6.5. Let p be a prime. We proved in Corollary 6.18 that a group with p2 elements must be abelian. Let G be a group with p3 elements. (a) Mimic the proof of Corollary 6.18 to try to prove that G is abelian. Where does the proof go wrong? (b) Give two examples of non-abelian groups with 23 elements. (This shows that the proof in (a) can’t work.) 6.6. Let G be a group, let X be a set, and let SX be the symmetry group of X as defined in Example 2.14. Let : G 􀀀! SX be a function from G to SX, and for g 2 G and x 2 X, let g  x = (g)(x). Prove that this defines a group action if and only if the function is a group homomorphism. Section 6.4. Sylow’s Theorem 6.7. Let p be prime, let n  1, let a  1, and consider the number A = a(a + 1)(a + 2)    (a + pn 􀀀 1): Factor A as A = pkB with p - B. (a) Prove that k does not depend on the value of a; cf. Lemma 6.20. (b) Find a simple closed formula for k in terms of p and n. 6.8. This exercise asks you to give two different proofs of the following stronger version of the first part of Sylow’s Theorem. Theorem: Let G be a finite group, let p be a prime, and suppose that #G is divisible by pr. Prove that G has a subgroup of order pr. (Note that pr is not required to be the largest power of p that divides G.) (b) Give a proof by induction on the power of p that divides#G. Use the fact that we already proved that G has a p-Sylow subgroup H. Then use Theorem 6.17 to deduce that H has a non-trivial center Z(H), and apply the induction hypothesis to H=Z(H). 6.9. This exercise descrbes a way to create new groups from known groups. Let G be a group. An isomorphism from G to itself is called an automorphism of G. The set of automorphisms is denoted Aut(G) = fgroup isomorphisms G ! Gg: We define a compostion law on Aut(G) as follows: for ; 2 Aut(G), we define to be the map from G to G given by ( )(g) = 􀀀 (g)  . (a) Prove that this composition law makes Aut(G) into a group. (b) Let a 2 G. Define a map a from G to G by the formula a : G 􀀀! G; a(g) = aga􀀀1: Prove that a 2 Aut(G), and that the map G 􀀀! Aut(G); a 7􀀀! a; (6.24) is a group homomorphism. (c) Prove that the kernel of homomorphism (6.24) is the center Z(G) of G. (d) Elements of Aut(G) that are equal to a for some a 2 G are called inner automorphisms, and all other elements of Aut(G) are called outer automorphisms. Prove that G is abelian if and only if its only inner automorphism is the identity map. 6.10. Let Cn be a cyclic subgroup of order n, and let Aut(Cn) be the automorphism group of Cn; see Exercise 6.9. Prove that Aut(Cn) is isomorphic to (Z=nZ), the group of units in the ring Z=nZ. Hint. Show that the following map is a group isomorphism: (Z=nZ) 􀀀! Aut(Cn); k mod n 7􀀀! k; where k(g) = gk. 6.11. In Example 6.24 we showed that there are exactly two groups of order 10. Do a similar calcuation to find all groups of order 15. 6.12. Let G be a finite group of order #G = pq, where p and q are primes satisfying p > q. Assume further that p 6 1 (mod q). (a) Prove that G is an abelian group. (Hint. Example 6.25 provides a starting point for the proof.) (b) Prove that G is a cyclic group.

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