Let \(G\) be a group and \(g \in G\) with \(\circ(g) =n_1n_2\) , where \(n_1\) and \(n_2\) are coprime positive integers. Then show that there are elements \(g_1,g_2 \in G\) such that \(g = g_1g_2 =g_2g_1\) and \(\circ(g_1) =n_1 , \circ(g_2) =n_2\) . Further show that \(g_1\) and \(g_2\) are uniquely determined by these conditions.

Let \(G\) be a group such that the intersection of all its subgroups which are different from \(\{e\}\) is a subgroup different from \(\{e\}\) . Prove that every element in \(G\) has finite order.

If a group \(G \neq \{e\}\) has no nontrivial subgroups, show that \(G\) must be a finite group of prime order.

Give an example of a group \(G\) having a subgroup \(H\) and two elements \(a,b\) such that \(Ha=Hb\) but \(aH \neq bH\) .

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 \(N_H(A) \le H\) .