Index of a subgroup

In mathematics, if G is a group, H a subgroup of G, and g an element of G, then

gH = { gh : h an element of H } is a left coset of H in G, and

Hg = { hg : h an element of H } is a right coset of H in G.

Some properties

We have gH = H if and only if g is an element of H. Any two left cosets are either identical or disjoint. The left cosets form a partition of G: every element of G belongs to one and only one left coset. The left cosets of H in G are the equivalence classes under the equivalence relation on G given by x ~ y if and only if x ^-1y ∈ H. Similar statements are also true for right cosets. A set of representatives of the cosets is called a transversal.

All left cosets and all right cosets have the same number of elements (or cardinality in the case of an infinite H). Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of H in G, written as [G : H]. The following useful formula often allows to compute the index:

|G| = [G : H] · |H|

(where |G| and |H| denote the cardinality of the two groups). See theorem of Lagrange for a proof.

Therefore, if G is a finite group, then the number of left cosets of H is equal to the order of G divided by the order of H. The same is true for the number of right cosets of H.

The subgroup H is normal if and only if gH = Hg for all g in G. In this case one can turn the set of all cosets into a group, the factor group of G by H.