# Tagged: subgroup criteria

## Problem 522

Let $G$ be an abelian group and let $H$ be the subset of $G$ consisting of all elements of $G$ of finite order. That is,
$H=\{ a\in G \mid \text{the order of a is finite}\}.$

Prove that $H$ is a subgroup of $G$.

## Problem 160

Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$.

Then show that $H$ is a subgroup of $G$.