Lagrange's theorem (group theory)

In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then ${\displaystyle |H|}$ is a divisor of ${\displaystyle |G|}$, i.e. the order (number of elements) of every subgroup H divides the order of group G.

The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup ${\displaystyle H}$ of a finite group ${\displaystyle G}$, not only is ${\displaystyle |G|/|H|}$ an integer, but its value is the index ${\displaystyle [G:H]}$, defined as the number of left cosets of ${\displaystyle H}$ in ${\displaystyle G}$.

This variant holds even if ${\displaystyle G}$ is infinite, provided that ${\displaystyle |G|}$, ${\displaystyle |H|}$, and ${\displaystyle [G:H]}$ are interpreted as cardinal numbers.