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 coset representative is a representative in the equivalence class sense. A set of representatives of all 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]. Lagrange's theorem allows us to compute the index in the case where G and H are finite, as per the formula:
- |G| = [G : H] · |H|
This equation also holds in the case where the groups are infinite (but is somewhat less useful).
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.