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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 mathematics, a product of groups usually refers to a direct product of groups, but may also mean: semidirect product; Product of group subsets; wreath product;
The Pauli group is the central product of the cyclic group and the dihedral group.; Every extra special group is a central product of extra special groups of order p 3.; The layer of a finite group, that is, the subgroup generated by all subnormal quasisimple subgroups, is a central product of quasisimple groups in the sense of Gorenstein.
In the special case of the category of groups, a product always exists: the underlying set of is the Cartesian product of the underlying sets of the , the group operation is componentwise multiplication, and the (homo)morphism : is the projection sending each tuple to its th coordinate.
The category-theoretical product in Grp is just the direct product of groups while the category-theoretical coproduct in Grp is the free product of groups. The zero objects in Grp are the trivial groups (consisting of just an identity element).
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a ...
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
A group in which every subgroup permutes is called an Iwasawa group. The subgroup lattice of an Iwasawa group is thus a modular lattice, so these groups are sometimes called modular groups [6] (although this latter term may have other meanings.) The assumption in the modular law for groups (as formulated above) that Q is a subgroup of S is