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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.
Kernel of a group homomorphism. It is the preimage of the identity in the codomain of a group homomorphism. Every normal subgroup is the kernel of a group homomorphism and vice versa. Direct product, direct sum, and semidirect product of groups. These are ways of combining groups to construct new groups; please refer to the corresponding links ...
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.
The direct product for modules (not to be confused with the tensor product) is very similar to the one that is defined for groups above by using the Cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components.
In mathematics, a product of groups usually refers to a direct product of groups, but may also mean: semidirect product; Product of group subsets;
The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups. Furthermore, every finite nilpotent group is the direct product of p-groups. [5] The multiplicative group of upper unitriangular n × n matrices over any field F is a nilpotent group of nilpotency class n − 1.
Consider the group G = S 3 × (the group of order 12 that is the direct product of the symmetric group of order 6 and a cyclic group of order 2). The center of G is isomorphic to its second factor Z 2 {\displaystyle \mathbb {Z} _{2}} .
The iterated wreath products of cyclic groups of order p are very important examples of p-groups. Denote the cyclic group of order p as W(1), and the wreath product of W(n) with W(1) as W(n + 1). Then W(n) is the Sylow p-subgroup of the symmetric group Sym(p n). Maximal p-subgroups of the general linear group GL(n,Q) are direct products of ...