If \(H\) is of finite index in \(G\), prove that there is a subgroup \(N\) of \(G\), contained in \(H\) and of finite index in \(G\) such that \(aNa^{-1}=N\) for all \(a \in G\) .

Let \(G\) be a finite group. Show that the only homomorphism from \(G\) to \((\mathbb{Z} ,+)(\) or to \((\mathbb{Q} ,+)\) or to \((\mathbb{R} ,+) )\) is the zero homomorphism.

Show that any endomorphism of \(( \mathbb{Q} ,+)\) is multiplication by a rational number.

 Prove that there is no non-trivial homomorphism from

(i) \((\mathbb{Q} ,+)\) to \((\mathbb{Z} ,+)\)

(ii) \((\mathbb{Q} ,+)\) to \((\mathbb{Z_m} ,+)\)

(iii) \((\mathbb{Q} ,+)\)to \((\mathbb{Q^\ast} ,\times)\), where \(\mathbb{Q^\ast} = \mathbb{Q} - \{0\}\).

Prove that there is no surjective homomorphism from

(i) \(\mathbb{Z_{16}} \times \mathbb{Z_{2}} \) to \(\mathbb{Z_{4}} \times \mathbb{Z_{4}} \)        (ii) \(\mathbb{Z_{4}} \times \mathbb{Z_{4}} \) to \(\mathbb{Z_8}\) .

If \(\phi\) is a homomorphism from \(\mathbb{Z_{30}}\) onto a group of order 5, determine ker \(\phi\) .

Let \(G\) be an abelian group and \(n\) a positive integer. Define \(f: G \rightarrow G\) by \(f(x) =x^n\) . Show that \(f\) is an endomorphism of \(G\). Further if \(G\) is finite such that \(n\) and \(|G|\) are coprime, show that \(f\) is an automorphism of \(G\) .

If a group \(G\) contains an element a  with exactly two conjugates, then show that \(G\) has a normal subgroup \(N \neq \{e\}\) .

Let \(H\) be a proper subgroup of a finite group \(G\) . Show that \(G\) contains at least one element that is not in any conjugate of \(H\) .

Let \(G\) be a non-abelian group of order 21. Prove that  \(Z(G)=\{e\}\) . Hence show that there is at most one possible class equation for \(G\).

Let \(G\) be an abelian group and \(H,K\) two finite subgroups of \(G\) of orders \(m\) and \(n\), respectively. Prove that \(G\) has a subgroup of order \(LCM(m,n)\) .