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:
“No direct hit to smash the shatter-proof
And lodge at last the quivering needle
Clean in the eye of one who stands transfixed
In fascination of her brightness.”
—Karl Shapiro (b. 1913)
“The history is always the same the product is always different and the history interests more than the product. More, that is, more. Yes. But if the product was not different the history which is the same would not be more interesting.”
—Gertrude Stein (1874–1946)
“... 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–?)