Category:Definitions/Index of Subgroups
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Let $G$ be a group.
Let $H$ be a subgroup of $G$.
If $G / H$ is a finite set, then $\index G H$ is finite, and $H$ is of finite index in $G$.
If $G / H$ is an infinite set, then $\index G H$ is infinite, and $H$ is of infinite index in $G$.