91Ó°ÊÓ

Let \(f: G \rightarrow G^{\prime}\) be an isomorphism of groups. Let \(a \in G\). Show that the period of \(a\) is the same as the period of \(f(a)\).

Let \(G\) be a group, and \(a, b, c\) be elements of \(G .\) If \(a b=a c\), show that \(b=c\).

Let \(V\) be a veetor space of dimension \(n\) over the field \(K .\) Show that. \(G L(n, K)\) is isomorphie to \(G L(V) .\)

Let \(G, G^{\prime}\) be finite groups of orders \(m, n\) respectively. What is the order of \(G \times G^{\prime} ?\)

Consider the additive group of integers \(\mathbf{Z}\). Show that it has only two generators, namely 1 and \(-1 .\) In general, show that an infinite cyclic group has only two generators.

Let \(f: G \rightarrow G^{\prime}\) be a homomorphism, and let \(H^{\prime}\) be a subgroup of \(G^{\prime} .\) Show that \(f^{-1}\left(H^{\prime}\right)\) is a subgroup of \(G\). If \(H^{\prime}\) is normal in \(G^{\prime}\), show that \(f^{-1}\left(H^{\prime}\right)\) is a normal subgroup of \(G\).

Show that the symmetrie group \(S_{3}\) is generated by the permutations $$ \left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 3 & 1 \end{array}\right] \text { and }\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 1 & 3 \end{array}\right] $$

Let \(G\) be a group. Let \(a\) be an element of \(G\). Let $$ \sigma_{a}: G \rightarrow G $$ be the map such that $$ \sigma_{a}(x)=a x a^{-1} $$ Show that the set of all such maps \(\sigma_{a}\) with \(a \in G\) is a group.

Let \(G\) be a finite abelian group of order \(n\), and let \(a_{1}, \ldots, a_{\mathrm{n}}\) be its elements. Show that the product \(a_{1} \cdots a_{n}\) is an element whose square is the unit element.

Let \(S_{3}\) be the symmetric group, and let \(\epsilon: S_{3} \rightarrow\\{1,-1\\}\) be the homomorphism given by the sign of the permutation. What is the order of the kernel of \(\epsilon ?\)