Direct Product of Groups - Definition

Definition

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

  1. 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.
  2. 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:

    “The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
    —William James (1842–1910)

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)