Show that if every cyclic subgroup of a group \(G\) is normal in \(G\), then every subgroup of \(G\) is normal.

Suppose that \(H\) is the only subgroup of order \(\mid H \mid \) in a finite group \(G\). Prove that \(H\) char \(G\), and hence  \(H \unlhd G\) .

Let \(H \le G \ and \ N(H) = \{g\in G \mid gHg^{-1}=H \}\). Prove that

(i) \(N(H) \le G\),    (ii) \(H \unlhd N(H)\),    (iii) If \(K \le G\) such that \(H \unlhd K\), then \(K \subseteq N(H)\),

(iv) \(H \unlhd G\) iff \(N(H)=G\).

If \(G\) is a finite group and \(N \unlhd G\) such that \(\mid N \mid \) and \([G:N]\) are coprime, show that for any \(a \in G ,a ^{\mid N \mid } = e \Rightarrow a \in N\).

If \(H \le G\), such that \(x^2 \in H\) for all \(x \in G\). Prove that \(H \unlhd G \) and \(G/H\) is abelian.

Show that if \(N\) is a normal subgroup of finite index in a group \(G\), and \(H\) is a subgroup of finite order in \(G\) such that \([G:N]\) and \(\mid H \mid \) are relatively prime, then \(H \subseteq N\).

Show that if \(N\) is a normal subgroup of finite order in a group \(G\), and \(H\) is a subgroup of finite index in \(G\) such that \([G:H]\) and \(\mid N \mid \) are relatively prime, then \(N \subseteq H\).

Show that if \(G\) is a group with centre \(Z(G)\), and if \(G/Z(G)\) is cyclic, then \(G\) is abelian. Hence show that there does not exist any group \(G\) such that \(\mid G/Z(G) \mid = p\), where \(p\) is a prime number.

Prove that \(\mathbb{Q/Z}\) is isomorphic to the multiplicative group of root of unity in \(\mathbb{C^\times}\).

Show that \((\mathbb{Q},+)\) has no proper subgroup of finite index. Deduce that \(\mathbb{Q/Z}\) has no proper subgroups of finite index.

Show that an infinite group is cyclic if and only if it is isomorphic to each of its nontrivial subgroups.