Central product

Last updated April 07, 2024

In mathematics, especially in the field of group theory, the central product is one way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroups of the smaller groups are merged into a single central subgroup of the product. Central products are an important construction and can be used for instance to classify extraspecial groups.

Definition

There are several related but distinct notions of central product. Similarly to the direct product, there are both internal and external characterizations, and additionally there are variations on how strictly the intersection of the factors is controlled.

A group G is an internal central product of two subgroups H, K if

G is generated by H and K.
Every element of H commutes with every element of K. ( Gorenstein 1980 , p. 29)

Sometimes the stricter requirement that $H\cap K$ is exactly equal to the center is imposed, as in ( Leedham-Green & McKay 2002 , p. 32). The subgroups H and K are then called central factors of G.

The external central product is constructed from two groups H and K, two subgroups $H_{1}\leq Z(H)$ and $K_{1}\leq Z(K)$ , and a group isomorphism $\theta \colon H_{1}\to K_{1}$ . The external central product is the quotient of the direct product $H\times K$ by the normal subgroup

N=\{(h,k):h\in H_{1},k\in K_{1},{\text{ and }}\theta (h)\cdot k=1\}

,

( Gorenstein 1980 , p. 29). Sometimes the stricter requirement that H₁ = Z(H) and K₁ = Z(K) is imposed, as in ( Leedham-Green & McKay 2002 , p. 32).

An internal central product is isomorphic to an external central product with H₁ = K₁ = H ∩ K and θ the identity. An external central product is an internal central product of the images of H × 1 and 1 × K in the quotient group $(H\times K)/N$ . This is shown for each definition in ( Gorenstein 1980 , p. 29) and ( Leedham-Green & McKay 2002 , pp. 32–33).

Note that the external central product is not in general determined by its factors H and K alone. The isomorphism type of the central product will depend on the isomorphism θ. It is however well defined in some notable situations, for example when H and K are both finite extra special groups and $H_{1}=Z(H)$ and $K_{1}=Z(K)$ .

Examples

The Pauli group is the central product of the cyclic group $C_{4}$ and the dihedral group $D_{4}$ .
Every extra special group is a central product of extra special groups of order p³.
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.

Applications

The representation theory of central products is very similar to the representation theory of direct products, and so is well understood, ( Gorenstein 1980 , Ch. 3.7).

Central products occur in many structural lemmas, such as ( Gorenstein 1980 , p. 350, Lemma 10.5.5) which is used in George Glauberman's result that finite groups admitting a Klein four group of fixed-point-free automorphisms are solvable.

In certain context of a tensor product of Lie modules (and other related structures), the automorphism group contains a central product of the automorphism groups of each factor ( Aranda-Orna 2022 , $\S$ 4).

Related Research Articles

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In abstract algebra, the center of a group $G$ is the set of elements that commute with every element of $G$ . It is denoted $Z(G)$ , from German Zentrum, meaning center. In set-builder notation,

In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.

In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.

In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pⁿ copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem.

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:

In mathematics, a group G is called the direct sum of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information. A group which can be expressed as a direct sum of non-trivial subgroups is called decomposable, and if a group cannot be expressed as such a direct sum then it is called indecomposable.

A group is a set together with an associative operation that admits an identity element and such that there exists an inverse for every element.

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If $and are two groups, then is an extension of by if there is a short exact sequence$

<span class="mw-page-title-main">Projective linear group</span> Construction in group theory

In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive.

In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers O_K factorise as products of prime ideals of O_L, provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of G need be considered, rather than two. This was certainly familiar before Hilbert.

In mathematics, especially in the area of algebra known as group theory, the Fitting subgroupF of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable. When G is not solvable, a similar role is played by the generalized Fitting subgroupF^*, which is generated by the Fitting subgroup and the components of G.

In mathematics, the binary icosahedral group 2I or ⟨2,3,5⟩ is a certain nonabelian group of order 120. It is an extension of the icosahedral group I or (2,3,5) of order 60 by the cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism

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 $G \times H$ . 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, an approximately finite-dimensional (AF) C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by Ola Bratteli. Later, George A. Elliott gave a complete classification of AF algebras using the K₀ functor whose range consists of ordered abelian groups with sufficiently nice order structure.

In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.

References

Gorenstein, Daniel (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 0569209
Leedham-Green, C. R.; McKay, Susan (2002), The structure of groups of prime power order, London Mathematical Society Monographs. New Series, vol. 27, Oxford University Press, ISBN 978-0-19-853548-5, MR 1918951
Aranda-Orna, Diego (2022), On the Faulkner construction for generalized Jordan superpairs, Linear Algebra and its Applications, vol. 646, pp. 1–28

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.