**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=S*_{3} 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.