coset

Given a subgroup H of a group G and an element a of G, the left coset aH is the subset of G whose elements can be written as a.h for some h in H. H and its left cosets partition G into equal subsets.

The aX notation can be extended to left cosets X. The notation abH is still unambiguous: a(bH) = { a.x | x in bH } = {a.b.h | h in H} = (ab)H. In effect, this defines a group on the left cosets of H.

Analogously, the right cosets of H are Ha = {h.a | h in H, a in G}. We can now consider how to interpret things like aHbHcHd. If the left and right cosets are the same, i.e. if for all a there is a b such that aH = Hb, then all of these are cosets as well; but this is not true in general. If aH = Ha for all a, we call H a normal divisor of G. In this case H and the coset group of H act as independent factors of G: G is their direct product.

Correction to rp's write-up.

If aH=Ha for all a in G then H is called a normal subgroup of G.

In general it is not true that just because H is a normal subgroup then G is (isomorphic to) the direct product of G and G/H. An example is the Symmetric group G=S₃ with normal subgroup H=<(123)>

Supplementary notes

If the group G is abelian we sometimes write its binary operation as addition (+) (for example, we do this for the integers Z). In that case we write the cosets additively too. So that a+H denotes the left coset {a+h: h in H}. Note that for abelian groups all left cosets are right cosets and so all subgroups are normal subgroups.