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 \(|G/Z(G)| \neq 77\) .

If \(G\) is a group with \(|G|=p^n\) , where \(p\) is a prime, and if \(0 \le k \le n\) , show that \(G\) contains a normal subgroup of order \(p^k\)  .

Let \(H\) be a proper subgroup of a finite p-group \(G\) . If \(H=p^s\) , then prove that there is a subgroup of \(G\) order \(p^{s+1}\) containing \(H\) .

Prove that a finite p-group cannot be simple unless it has order p

Let G be a finite group and p a prime. Show that a normal p-subgroup of G is contained in every Sylow p-subgroup of G.