# Difference between revisions of "Subgroup, index of a"

in a group $G$
The number of cosets (cf. Coset in a group) in any decomposition of $G$ with respect to this subgroup $H$ (in the infinite case, the cardinality of the set of these cosets). If the number of cosets is finite, $H$ is called a subgroup of finite index in $G$. The intersection of a finite number of subgroups of finite index itself has finite index (Poincaré's theorem). The index of a subgroup $H$ in $G$ is usually denoted by $\left|G:H\right|$. The product of the order of a subgroup $H$ by its index $\left|G:H\right|$ is equal to the order of $G$ (Lagrange's theorem). This relationship applies to finite groups $G$ and also to infinite groups $G$ for the corresponding cardinalities.