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.