Index (Groups)

index

For $H \leq G$ (see Subgroup) the index of $H$ in $G$ is the number of cosets of $H$ in $G$ .

This is often denoted $[G \cdot H]$ or $| G : H |$ . Notice that this just $| \frac{G}{H} | = [G \cdot H]$ by definition. If $G$ is finite ( $| G | < \infty$ ) then $[G \cdot H] = \frac{| G |}{| H |}$ via Lagrange's Divisibility Theorem of Order of Subgroups.