A subgroup *N* of a group *G* is a subset of the set of elements of *G* (in
which the group relations still hold.) If we multiply the elements of *N* by
an element *a* not in *N* we get a set of products, distinct from *N*, and of
the same size as *N*. This set is called a coset. If there are elements not
in *N* or the coset we can multiply *N* by one of those and get a new coset,
distinct from the first two sets. In general it matters which side we
multiply on:
.
If *aN* = *Na*, (meaning not that individual
products are equal, but that the set of products are equal), then *N* is
called normal or invariant. This is equivalent to being self-conjugating,
where
*g*^{-1}*Ng* = *N*, i.e.
*g*^{-1}*n*_{1}*g* = *n*_{2} where *n*_{1} and *n*_{2} are
members of *N*. (See bottom of I)

Illustration: given *Dih*_{3}, one subgroup is *I*, *s*, *s*^{2}. We can multiply on
the right by *t*, which is not in the subgroup, and get
*t*, *st*, *s*^{2}*t*=*ts*. Or
on the left to get
*t*, *ts*, *ts*^{2}=*st*. In this case it's the same coset,
permuted a bit. If we multiply by *st* we get the same set:
*st*, *s*^{t}=*ts*,
*s*^{3}*t*=*t*. On the other hand, we can consider the subgroup *I*,*t* and multiply
by *s* to get *s*, *ts* on right and *s*, *st* on the left. Not the same cosets.
So *I*, *s*, *s*^{2} is a normal subgroup of *Dih*_{3}, with a coset of
*t*, *st*, *s*^{2}*t*,
while *I*, *t* (or *I*, *l* and *I*, *r*) isn't a normal subgroup.

We could also have looked at conjugation, e.g.
*tIt*=*I*, *tst*=*s*^{2}, *ts*^{2}*t*=*s*, and
this is true for conjugation by any element in *Dih*_{3}.