Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted ${\displaystyle |G:H|}$ or ${\displaystyle [G:H]}$ or ${\displaystyle (G:H)}$. Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula

${\displaystyle |G|=|G:H||H|}$

(interpret the quantities as cardinal numbers if some of them are infinite). Thus the index ${\displaystyle |G:H|}$ measures the "relative sizes" of G and H.

For example, let ${\displaystyle G=\mathbb {Z} }$ be the group of integers under addition, and let ${\displaystyle H=2\mathbb {Z} }$ be the subgroup consisting of the even integers. Then ${\displaystyle 2\mathbb {Z} }$ has two cosets in ${\displaystyle \mathbb {Z} }$, namely the set of even integers and the set of odd integers, so the index ${\displaystyle |\mathbb {Z} :2\mathbb {Z} |}$ is 2. More generally, ${\displaystyle |\mathbb {Z} :n\mathbb {Z} |=n}$ for any positive integer n.

When G is finite, the formula may be written as ${\displaystyle |G:H|=|G|/|H|}$, and it implies Lagrange's theorem that ${\displaystyle |H|}$ divides ${\displaystyle |G|}$.

When G is infinite, ${\displaystyle |G:H|}$ is a nonzero cardinal number that may be finite or infinite. For example, ${\displaystyle |\mathbb {Z} :2\mathbb {Z} |=2}$, but ${\displaystyle |\mathbb {R} :\mathbb {Z} |}$ is infinite.

If N is a normal subgroup of G, then ${\displaystyle |G:N|}$ is equal to the order of the quotient group ${\displaystyle G/N}$, since the underlying set of ${\displaystyle G/N}$ is the set of cosets of N in G.