91Ó°ÊÓ

Let \(G\) and \(H\) be finite groups. If there are normal series of \(G\) and of \(H\) having the same set of factor groups, then \(G\) and \(H\) have the same composition factors.

Prove that the dihedral groups \(D_{2 n}\) are solvable.

Burnside proved (using Representation Theory) that the number of elements in a conjugacy class of a finite simple group can never be a prime power larger than 1. Use this fact to prove Buruside's theorem: If \(p\) and \(q\) are primes, then every group of order \(p^{m} q^{n}\) is solvable.

Show that the following conditions on a finite group \(G\) are equivalent: (i) \(G\) is nilpotent; (ii) \(G\) satisfies the normalizer condition; (iii) Every maximal subgroup of \(G\) is normal.

If \(G\) is a nilpotent group and \(H\) is a minimal normal subgroup of \(G\), then \(H \leq Z(G)\)

The dihedral group \(D_{2 n}\) is nilpotent if and only if \(n\) is a power of \(2 .\)

Let \(G\) be a finite nilpotent group of order \(n\). If \(m \mid n\), then \(G\) has a subgroup of order \(m\).

(i) Show \(\gamma_{i}\left(\mathrm{UT}\left(n, \mathbb{Z}_{g}\right)\right)\) consists of all upper triangular matrices with I's on the main diagonal and 0 's on the \(i-1\) superdiagonals just above the main diagonal (Hint. If \(A\) is unitriangular, consider powers of \(A-E\), where \(E\) is the identity matrix.) (ii) The group \(\mathrm{UT}\left(n, \mathbb{Z}_{p}\right)\) of all \(n \times n\) unitriangular matrices over \(\mathbb{Z}_{p}\) is a \(p\) -group that is nilpotent of class \(n-1\).

(Wielandt). A finite group \(G\) is nilpotent if and only if \(G^{\prime} \leq \Phi(G)\).