Prove that the homomorphic image of a cyclic (abelian) group is cyclic (abelian). Give an example to show that if the homomorphic image of a group is cyclic (abelian), the same may not hold for the group itself.

In \((\mathbb{Z_{24}},+)\) find all \(m \in \mathbb{Z_{24}}\) such that \(\langle 21 \rangle = \langle m \rangle \).

Show that \((\mathbb{Q^+}, .)\), the multiplicative group of positive rational numbers, is not a cyclic group.

Let \(f:G \rightarrow G^\prime \) be a homomorphism of groups. Show that if \(H^\prime \unlhd G^\prime\) , then \(f^{-1}(H^\prime ) \unlhd G\) and \(\ker f \subseteq f^{-1}(H^\prime)\). Further show that if \(f\) is surjective and \(H \unlhd G \), then \(f(H) \unlhd G^\prime \) .

Show by an example that we can find groups \(G,H,K\) such that \(K \unlhd G\) and \(H \unlhd K\), but \(H\) is not a normal subgroup of \(G\).

Give an example of a group \(G\) and a subgroup \(H\) of \(G\) such that for some \(a \in G , aHa^{-1} \subsetneq H\).

If a cyclic subgroup \(T\) of a group \(G\) is normal in \(G\), then show that every subgroup of \(T\) is normal in \(G\).

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 \(a \in G ,a ^{\mid N \mid } = e \Rightarrow a \in N\).