Let \(H \le G\), and \(a \in G\) . Let \(aHa^{-1} = \{aha^{-1} |h \in H \}\) . Show that \(aHa^{-1}\) is a subgroup of \(G\).

Suppose that \(H \le G\) such that whenever \(Ha \neq Hb\) then \(aH \neq bH\) . Prove that \(gHg^{-1} \subseteq H\) for all \(g \in G\) .

If \(H\) and \(K\) are subgroups of orders \(p\) and \(n\) , respectively, where \(p\) is a prime number, show that either \(H \cap K=\{e\}\) or \(H \le K\) .

If \(G\) is a group and \(K \le H \le G \) , then show that \([G:K] =[G:H][H:K]\) (assume finite indices). Hence deduce that if \([G:H] =p\), where \(p\) is a prime, then either \(H=K\) or \(H=G\) .

If \(G\) is a group and \(H,K\) are two subgroups of finite indices in \(G\), prove that \(H \cap K\) is of finite index in \(G\). Can you find an upper bound for the index of \(H \cap K\) in \(G\)?

If \(H\) and \(K\) are subgroups of finite indices of a group \(G\) such that \([G:H]\) and \([G:K]\) are relatively prime, show that \(G =HK\) .

For any subgroup \(H\) of \(G\) and any non-empty subset \(A\) of \(G\) define \(N_H(A) = \{ h \in H | hAh^{-1} = A \}\) . Show that \(N_H(A) =N_G(A) \cap H\) and dedue that \(N_H(A) \le H\) .

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.