Index of $H$ in $G$, or $|G:H|$

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Definition (Index of $H$ in $G$, or $|G:H|$)

If $H$ is a subgroup of $G$, then the index of $H$ in $G$, written $|G:H|$, is the number of distinct right (or left) cosets of $H$. Because of Lagrange's theorem, we know that $|G:H|=|G|/|H|$.

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