Quotient Group - Product of Subsets of A Group

Product of Subsets of A Group

In the following discussion, we will use a binary operation on the subsets of G: if two subsets S and T of G are given, we define their product as ST = {st : s ∈ S ∧ t ∈ T}. This operation is associative and has as identity element the singleton {e}, where e is the identity element of G. Thus, the set of all subsets of G forms a monoid under this operation.

In terms of this operation we can first explain what a quotient group is, and then explain what a normal subgroup is:

A quotient group of a group G is a partition of G which is itself a group under this operation.

It is fully determined by the subset containing e. A normal subgroup of G is the set containing e in any such partition. The subsets in the partition are the cosets of this normal subgroup.

A subgroup N of a group G is normal if and only if the coset equality aN = Na holds for all a in G. In terms of the binary operation on subsets defined above, a normal subgroup of G is a subgroup that commutes with every subset of G and is denoted N ◁ G. A subgroup that permutes with every subgroup of G is called a permutable subgroup.