Direct Product Of Groups
In the mathematical field of 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 fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.
Read more about Direct Product Of Groups: Definition, Examples, Elementary Properties, Algebraic Structure
Famous quotes containing the words direct, product and/or groups:
“Irony, forsooth! Guard yourself, Engineer, from the sort of irony that thrives up here; guard yourself altogether from taking on their mental attitude! Where irony is not a direct and classic device of oratory, not for a moment equivocal to a healthy mind, it makes for depravity, it becomes a drawback to civilization, an unclean traffic with the forces of reaction, vice and materialism.”
—Thomas Mann (1875–1955)
“In fast-moving, progress-conscious America, the consumer expects to be dizzied by progress. If he could completely understand advertising jargon he would be badly disappointed. The half-intelligibility which we expect, or even hope, to find in the latest product language personally reassures each of us that progress is being made: that the pace exceeds our ability to follow.”
—Daniel J. Boorstin (b. 1914)
“... until both employers’ and workers’ groups assume responsibility for chastising their own recalcitrant children, they can vainly bay the moon about “ignorant” and “unfair” public criticism. Moreover, their failure to impose voluntarily upon their own groups codes of decency and honor will result in more and more necessity for government control.”
—Mary Barnett Gilson (1877–?)