Advanced Algebra I
Homework 3 due on Oct.17, 2003 Part A.
(1) Find a example of a solvable group which is not nilpotent.
(2) Let G be a nilpotent group. Show that every subgroups and every quotient groups are nilpotent.
(3) Let G be a nilpotent group of order n. Prove that for m|n, there is a subgroup of order m in G.
(4) Let G be a finite solvable group. Show that G has a solv- able series such that each factor is a cyclic group of prime or- der. (Hint: you might need the fundamental theorem for finite abelian group).
Part B.
(1) Show that a group of order p2q , where p, q are distinct primes, is solvable.
(2) Show that a simple group of order 60 is isomorphic to A5. (Hint:
Consider the action G × G/H → G/H which gives a homomor- phims ˜α : G → S|G/H|. Show that this is an injection. And show that there is a subgroup H of index 5.)
(3) If H, K are solvable subgroups of G, and H C G, then HK is a solvable subgroup of G.
(4) Let G be a finite group and let N C G be a normal subgroup such that (|N|, |G/N|) = 1
(a) Let H be a subgroup that |H| = |G/N|. Prove that G = HN .
(b) Let σAut(G) be an automorphism. Prove that σ(N) = N.
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