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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.
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 center of the symmetric group, S n, is trivial for n ≥ 3. The center of the alternating group, A n, is trivial for n ≥ 4. The center of the general linear group over a field F, GL n (F), is the collection of scalar matrices, { sI n ∣ s ∈ F \ {0} }. The center of the orthogonal group, O n (F) is {I n, −I n}.
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;
A basic fact about perfect groups is Otto Grün's proposition of Grün's lemma (Grün 1935, Satz 4, [note 1] p. 3): the quotient of a perfect group by its center is centerless (has trivial center). Proof: If G is a perfect group, let Z 1 and Z 2 denote the first two terms of the upper central series of G (i.e., Z 1 is the center of G , and Z 2 ...
The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
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}} .
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