Definition
Given groups G and H, the direct product G × H is defined as follows:
- The elements of G × H are ordered pairs (g, h), where g ∈ G and h ∈ H. That is, the set of elements of G × H is the Cartesian product of the sets G and H.
- The binary operation on G × H is defined componentwise:
(g1, h1) · (g2, h2) = (g1 · g2, h1 · h2)
The resulting algebraic object satisfies the axioms for a group. Specifically:
- Associativity
- The binary operation on G × H is indeed 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.
Read more about this topic: Direct Product Of Groups
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)