Let \(G\) be an abelian group and \(n\) a positive integer. Define \(f: G \rightarrow G\) by \(f(x) =x^n\) . Show that \(f\) is an endomorphism of \(G\). Further if \(G\) is finite such that \(n\) and \(|G|\) are coprime, show that \(f\) is an automorphism of \(G\) .

If a group \(G\) contains an element a  with exactly two conjugates, then show that \(G\) has a normal subgroup \(N \neq \{e\}\) .

Let \(H\) be a proper subgroup of a finite group \(G\) . Show that \(G\) contains at least one element that is not in any conjugate of \(H\) .

Let \(G\) be a non-abelian group of order 21. Prove that  \(Z(G)=\{e\}\) . Hence show that there is at most one possible class equation for \(G\).

Let \(G\) be an abelian group and \(H,K\) two finite subgroups of \(G\) of orders \(m\) and \(n\), respectively. Prove that \(G\) has a subgroup of order \(LCM(m,n)\) .

(a) Show that the center of a group \(G\) is a characteristic subgroup of \(G\).

(b) Prove that characteristic subgroups are normal. Give an example of a normal subgroup which is not characteristic.

(c) Show that if \(H\) is the only subgroup of order \(n\) in \(G\), or if \(H\) is the only subgroup of index \(k\) in \(G\), then \(H\) is characteristic in \(G\).

Prove that there are no simple groups of orders 42, 56, 96, 108, 200, 1986.

Let \(G\) be a group of order \(1575\). Prove that if \(H\) is a normal subgroup of order 9 in \(G\), then \(H \le Z(G)\).

Show that if \(|G| =pqr\) , where \(p,q,r\) are distinct primes, then \(G\) is not simple.

If a group of order \(p^n\) , where \(p\) is a prime, contains exactly one subgroup each of orders \(p,p^2, \cdots ,p^{n-1}\) , prove that it is cyclic.

Let \(G\) be a group. Prove that \(|G/Z(G)| \neq 77\) .