Product of group subsets

In mathematics, one can define a product of group subsets in a natural way. If S and T are subsets of a group G, then their product is the subset of G defined by

Note that S and T need not be subgroups for this product to be well defined. The associativity of this product follows from that of the group product. The product of group subsets therefore defines a natural monoid structure on the power set of G.

A lot more can be said in the case where S and T are subgroups.

Product of subgroups

If S and T are subgroups of G their product need not be a subgroup (consider, for example, two distinct subgroups of order two in the symmetric group on 3 symbols). This product is sometimes called the Frobenius product. In general, the product of two subgroups S and T is a subgroup if and only if ST = TS, and the two subgroups are said to permute. (Walter Ledermann has called this fact the Product Theorem, but this name, just like "Frobenius product" is by no means standard.) In this case, ST is the group generated by S and T, i.e. ST = TS = ⟨S ∪ T⟩.

Direct product of groups

In group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G ⊕ H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.

Definition

Given groups G and H, the direct product G × H is defined as follows:

The resulting algebraic object satisfies the axioms for a group. Specifically: