Group Theory Prelim Question Set 2 1. Let p be a prime

Group Theory Prelim Question Set 2 1. Let p be a prime number. Show that a subgroup of Sp that contains an element of order p and a transposition must be the whole of Sp. 2. Let H be a subgroup of the group G. Show that the following are equivalent: (a) x-1y1xy ? H for all x y ? G (b) H ? G and G/H is abelian. 3. Find two elements of S7 that have the same order but are not conjugate. Let p ? S7 be of maximal order. What is |p|? Is p an even permutation? If |?| = |pi| is ? conjugate to p? 4. Let G be a finite group and H a non-normal subgroup of G of index n > 1. Show that if |H| is divisible by a prime p = n then H can not be a simple group. 5. Let G be an abelian group. Let K = {a ? G : a2 = 1} and let H = x2 : xG. Show that G/K ~= H. 6. Let G be a group acting on S. H = G with the inherited action of H on S transitive. Show that ?t ? S(G = HGt). 7. Let G be a group. A subgroup H of G is called a characteristic subgroup of G if ?(H) = H for every automorphism ? of G. Show that if H is a characteristic subgroup of N and N is a normal subgroup of G then H is a normal subgroup of G. 1
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