Cosets and Normal Subgroups

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: $aN \not= Na$. 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^-1Ng = N, i.e. g^-1n₁g = n₂ where n₁ and n₂ are members of N. (See bottom of I)

Illustration: given Dih₃, one subgroup is I, s, s². We can multiply on the right by t, which is not in the subgroup, and get t, st, s²t=ts. Or on the left to get t, ts, ts²=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, s³t=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, s² is a normal subgroup of Dih₃, with a coset of t, st, s²t, 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=s², ts²t=s, and this is true for conjugation by any element in Dih₃.