Cosets

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

Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup. Although derived from a subgroup, cosets are not usually themselves subgroups of G, only subsets.

A coset is a left or right coset of some subgroup in G. Since Hg = g ( g⁻¹Hg ), the right coset Hg (of H with respect to g) and the left coset g ( g⁻¹Hg ) (of the conjugate subgroup g⁻¹Hg ) are the same. Hence it is not meaningful to speak of a coset as being left or right unless one first specifies the underlying subgroup. In other words: a right coset of one subgroup equals a left coset of a different (conjugate) subgroup. If the left cosets and right cosets are the same, then H is a normal subgroup and the cosets form a group called the quotient or factor group.

The map gH ↦ (gH)⁻¹ = Hg⁻¹ defines a bijection between the left cosets and the right cosets of H, so the number of left cosets is equal to the number of right cosets. The common value is called the index of H in G.

For abelian groups, left cosets and right cosets are always the same. If the group operation is written additively, the notation used changes to g + H or H + g.

Cosets are a basic tool in the study of groups; for example they play a central role in Lagrange's theorem.

Let G = ({−1,1}, ×) be the group formed by {−1,1} under multiplication, which is isomorphic to C₂, and H the trivial subgroup ({1}, ×). Then {−1} = (−1)H = H(−1) and {1} = 1H = H1 are the only cosets of H in G. Because its left and right cosets with respect to any element of G coincide, H is a normal subgroup of G.

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