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.

Which of the following Harappan city sites had bipartite division-high citadel and lower town?

(i) Harappa

(ii) Mohenjodaro

(iii) Kalibangan

(iv) SurkotadaSelect the correct answer from the codes given below:

Codes:

(a) (i) only                                 (b) (i) and (ii)

(c) (i), (ii) and (iii)                      (d) (i), (ii), (iii) and (iv)