Chapter 10

Let \(G\) be the symmetric group \(S(n)\) and \(V\) be a complex vector space, with basis \(\left\\{v_{1}, v_{2}, \ldots, v_{n}\right\\}\). For \(\pi \in G\) and any \(v=\lambda_{1} v_{1}+\cdots+\lambda_{n} v_{n}\) of \(V\), define $$ \pi \cdot v=\lambda_{1} v_{\pi(1)}+\cdots+\lambda_{n} v_{\pi(n)} $$ Show that \(V\) is a \(G\)-set, and find both \(\operatorname{orb}(v)\) and \(G_{v}\) when (a) \(n=4\) and \(v=v_{1}+v_{2}+v_{3}+v_{4} ;\) (b) \(n=4\) and \(v=v_{1}+v_{3}\).

Let \(X\) be a \(G\)-set and \(x \in X\). Show that for any \(g \in G\), the stabiliser of \(g: x\) is the subgroup \(g G_{x} g^{-1}\)

Determine the list of conjugacy classes in the symmetric group \(S(4)\) and also in the alternating group \(A(4)\).

Let \(G\) be any group and \(g\) be an element of \(G\). Prove directly that, \(C_{G}(g)=\\{x \in G: g x=x g\\}\) is a subgroup of \(G\).

Let \(G\) be a finite group with precisely two conjugacy classes. Prove that. \(G\) has two elements.

Let \(G\) be the group \(S(3) .\) Calculate \(N_{G}(H)\) when \(H\) is (a) the subgroup \(\\{1,(123),(132)\\}\) and (b) the subgroup \(\\{1,(12)\\}\).