## Union of Two Subgroups is Not a Group

## Problem 625

Let $G$ be a group and let $H_1, H_2$ be subgroups of $G$ such that $H_1 \not \subset H_2$ and $H_2 \not \subset H_1$.

**(a)** Prove that the union $H_1 \cup H_2$ is never a subgroup in $G$.

**(b)** Prove that a group cannot be written as the union of two proper subgroups.