Let \(h \le G\) . Let \(N = \bigcap_{x \in G} xHx^{-1}\) . Prove that \(N \le G\), and \(aNa^{-1} =N\) for all \(a \in G\) .

 If \(H\) is a subgroup of finite index in \(G\) , prove that there is only a finite number of distinct subgroups in \(G\) of the form \(aHa^ {-1}\) .

 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 \(H\) .