Chapter 18

Show that the class of nilpotent groups is not closed under extensions: there is a non-nilpotent group \(G\) with a normal subgroup \(N\) such that \(N\) and \(G / N\) are both nilpotent.

Let \(G\) be the group of order 27 consisting of the \(3 \times 3\) matrices of the form $$ A=\left(\begin{array}{lll} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array}\right) $$ where \(a, b\) and \(c\) are integers modulo 3, so that each of \(a, b\) and \(c\) may be taken to be one of 0,1 or 2 with arithmetical operations modulo 3 (so that, for example, \(1+2=0\) and \(2 \times 2=1\) ). Prove that every element of \(G\) has order 3 . Find the centre of \(G\). Hence find a chief series for \(G\).

Let \(P\) be the group $$ \left\langle x, y: x^{9}=y^{3}=1, y x y^{-1}=x^{4}\right\rangle $$ so that \(P\) consists of the 27 elements of the form \(x^{j} y^{k}\) with \(j\) being one of \(0,1, \ldots, 8\) and \(k\) being 0,1 or 2 . Show that \(\langle x\rangle\) is a normal subgroup of \(P\) and that \(x^{3}\) is in the centre of \(P\). Find a chief series for \(P\).

Show that groups of all possible orders between 61 and 71 (inclusive) are soluble.