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.