Math 594. Solutions for Problem Set 2
2.1. In the lecture notes, it was proven that there is some p-Sylow subgroup H of G such
that (H ) = . It follows that 1 () = H Z (G). This implies that H is normal in
1 (), and hence that H is the only p-Sylow subgr
Math 594. Solutions for Problem Set 1
1.1. (a) First, the set S is nite, since it is a union of nite sets Si of which only nitely
many are non-empty. Also, each group G is a p-group, so the calculation |G| = S |G |
shows that the order of G is a nite prod
Math 594 Assignment 3
Charles Stibitz
3.1. Supply a detailed proof of the following claim mentioned in class: if p is a given prime, then
an abelian group A such that p x = 0 for all x A is the same thing as a vector space over the
nite eld Fp with p elem
Math 594 Assignment 4
Charles Stibitz
=
4.1. Let A be an abelian group and let : A A be a group automorphism. Dene an action of
Z on A by the formula (n, a) n (a), where a A, n Z and n is the n-fold composition of
(so 0 = idA and n = (1 )n when n < 0). I
Math 594 Assignment 6
Charles Stibitz
6.1. Let G be a nite group. Prove that G is abelian if and only if every irreducible representation
of G is 1-dimensional.
First suppose that G is abelian. Let : G GL(V ) be an irreducible representation. Then we
have
Math 594 Assignment 5
Charles Stibitz
5.1. Let n 2 and let Dn denote the dihedral group of order 2n, i.e., the group of symmetries (rigid
motions) of a regular polygon with n vertices. For this problem you may assume that Dn consists
of the following elem
Math 594 Assignment 7
Charles Stibitz
7.1. Prove that the number of 1-dimensional representation of G (up to isomorphism) is equal to
|G/[G, G]| without using the classication of nite abelian groups.
First we reduce this to the abelian case. Suppose we ha