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 $|G:H|$ or $[G:H]$ or $(G:H)$ . 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

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

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

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

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

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

If H is a subgroup of G and K is a subgroup of H, then

The $A_{n}$ has index 2 in the symmetric group $S_{n},$ and thus is normal.

alternating group

The $\operatorname {SO} (n)$ has index 2 in the orthogonal group $\operatorname {O} (n)$ , and thus is normal.

special orthogonal group

The $\mathbb {Z} \oplus \mathbb {Z}$ has three subgroups of index 2, namely

free abelian group

Infinite index[edit]

If H has an infinite number of cosets in G, then the index of H in G is said to be infinite. In this case, the index $|G:H|$ is actually a cardinal number. For example, the index of H in G may be countable or uncountable, depending on whether H has a countable number of cosets in G. Note that the index of H is at most the order of G, which is realized for the trivial subgroup, or in fact any subgroup H of infinite cardinality less than that of G.

E^p(G) is the intersection of all index p normal subgroups; G/E^p(G) is an , and is the largest elementary abelian p-group onto which G surjects.

elementary abelian group

A^p(G) is the intersection of all normal subgroups K such that G/K is an abelian p-group (i.e., K is an index $p^{k}$ normal subgroup that contains the derived group $[G,G]$ ): G/A^p(G) is the largest abelian p-group (not necessarily elementary) onto which G surjects.

O^p(G) is the intersection of all normal subgroups K of G such that G/K is a (possibly non-abelian) p-group (i.e., K is an index $p^{k}$ normal subgroup): G/O^p(G) is the largest p-group (not necessarily abelian) onto which G surjects. O^p(G) is also known as the p-residual subgroup.

Index of a subgroup

alternating group

special orthogonal group

free abelian group

Infinite index[edit]

elementary abelian group

Virtually

Codimension

Normality of subgroups of prime index

Subgroup of least prime index is normal