Proper Subgroup

Let $G$ be a group . A proper subgroup of $G$ is a subgroup $H\le G$ such that $H\ne G$, equivalently $H$ is a proper subset of $G$.

Proper subgroups capture “nontrivial internal structure” of a group: if $G$ has no nontrivial normal proper subgroups, then $G$ is simple.

Examples:

• $2\mathbb{Z}$ is a proper subgroup of $\mathbb{Z}$.
• $A_n$ is a proper subgroup of $S_n$ for $n\ge 2$.
• The trivial subgroup $\{e\}$ is proper whenever $G$ is not the trivial group.