91Ó°ÊÓ

Show that the symmetric group \(S(n)\) is soluble for \(n \leq 4\) but is not soluble for \(n \geq 5\).

Let \(\vartheta: G \rightarrow H\) be a group homomorphism. Prove that (a) \((\vartheta(G))^{\prime}=\vartheta\left(G^{\prime}\right)\); (b) if \(H\) is soluble and \(\vartheta\) is injective then \(G\) is soluble; and (c) if \(G\) is soluble and \(\vartheta\) is surjective then \(H\) is soluble.

Let \(G\) be the set of maps from \(\mathbf{R}\) to \(\mathbf{R}\) of the form \(\vartheta_{a, b}\) (with \(a, b \in \mathbf{R}\) and \(a \neq 0\) ) where \(\vartheta_{n, b}(x)=a x+b .\) Show that \(G\) is a group under composition of maps and that the subset \(\left\\{\vartheta_{1, e}: e \in \mathbf{R}\right\\}\) is an abelian subgroup of \(G . \mathrm{By}\) considering commutators, or otherwise, deduce that \(G\) is a soluble group.

For any elements \(x, y\) of a group \(G\) denote by \(x^{y}\) the product \(y x y^{-1}\). Let \(a, b\) and \(c\) be elements of a group \(G\). Prove that (a) \([a b, c]=[b, c]^{a}[a, c]\); and (b) \([a, b c]=[a, b][a, c]^{b} .\)

Let \(G\) be the set of all real \(4 \times 4\) matrices of the form $$ \left(\begin{array}{llll} 1 & a & b & c \\ 0 & 1 & 0 & d \\ 0 & 0 & 1 & \mathrm{e} \\ 0 & 0 & 0 & 1 \end{array}\right) $$ Find a formula for the product of two elements of \(G\) and find the inverse of an elernent of \(G .\) Deduce that \(G\) is a group with respect to matrix multiplication. Let \(A\) be the subset of matrices in \(G\) of the form $$ \left(\begin{array}{llll} 1 & 0 & 0 & c \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$ Show that \(A\) is a normal abelian subgroup of \(G\). Prove that, \(G^{\prime}\) is contained. in \(A\) and deduce that \(G\) is soluble.