**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.

Read more about this topic: Quotient Group

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