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Concept
Direct product of groups
Summary
In mathematics, specifically 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 . 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.
Given groups G (with operation *) and H (with operation ∆), the direct product G × H is defined as follows:
The resulting algebraic object satisfies the axioms for a group. Specifically:
Associativity The binary operation on G × H is associative.
Identity The direct product has an identity element, namely (1G, 1H), where 1G is the identity element of G and 1H is the identity element of H.
Inverses The inverse of an element (g, h) of G × H is the pair (g−1, h−1), where g−1 is the inverse of g in G, and h−1 is the inverse of h in H.
Let R be the group of real numbers under addition. Then the direct product R × R is the group of all two-component vectors (x, y) under the operation of vector addition:
(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2).
Let R+ be the group of positive real numbers under multiplication. Then the direct product R+ × R+ is the group of all vectors in the first quadrant under the operation of component-wise multiplication
(x1, y1) × (x2, y2) = (x1 × x2, y1 × y2).
Let G and H be cyclic groups with two elements each:
Then the direct product G × H is isomorphic to the Klein four-group:
Let G and H be groups, let P = G × H, and consider the following two subsets of P:
G′ = { (g, 1) : g ∈ G } and H′ = { (1, h) : h ∈ H } .
Both of these are in fact subgroups of P, the first being isomorphic to G, and the second being isomorphic to H.
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