central product of groups


1 Definitions

A central decomposition is a set ℋ of subgroupsMathworldPlanetmathPlanetmath of a group G where

  1. 1.

    for 𝒥⊆ℋ, G=⟨𝒥⟩ if, and only if, 𝒥=ℋ, and

  2. 2.

    [H,⟨ℋ-{H}⟩]=1 for all H∈ℋ.

A group G is centrally indecomposableMathworldPlanetmath if its only central decomposition is {G}. A central decomposition is fully refined if its members are centrally indecomposable.

Remark 1.

Condition 1 is often relaxed to G=⟨H⟩ but this has the negative affect of allowing, for example, R2 to have the central decomposition such sets as {⟨(1,0)⟩,⟨(0,1)⟩,⟨(1,1)⟩,0,R2} and in general a decomposition of any possible size. By impossing 1, we then restrict the central decompositions of R2 to direct decompositions. Furthermore, with condition 1, the meaning of indecomposable is easily had.

A central product is a group G=(∏H∈ℋH)/N where N is a normal subgroupMathworldPlanetmath of ∏H∈ℋH and H∩N=1 for all H∈ℋ.

Proposition 2.

Every finite central decomposition H is a central product of the members in H.

Proof.

Suppose that ℋ is a a finite central decomposition of G. Then define π:∏H∈ℋH→G by (xH:H∈ℋ)↦∏H∈ℋxH. Then G=(∏H∈ℋH)/ker⁡π. Furthermore, H∩ker⁡π=1 for each direct factorMathworldPlanetmath H of ∏K∈ℋK. Thus, G is a central product of ℋ. ∎

2 Examples

  1. 1.

    Every direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is also a central product and so also every direct decomposition is a central decomposition. The converse is generally false.

  2. 2.

    Let E={[1αγ01β001]:α,β,γ∈K}, for a field K. Then G is a centrally indecomposable group.

  3. 3.

    If

    F={[1α1…αnγ010β1⋮⋱0⋮1βn1]:αi,βi,γ∈K,1≤i≤n}

    and

    Hi={A∈F:αj=0=βj⁢∀j≠i,1≤j≤n}

    then {H1,…,Hn} is a central decomposition of G. Furthermore, each Hi is isomorphicPlanetmathPlanetmathPlanetmath to E and so ℋ is a fully refined central decomposition.

  4. 4.

    If D8=⟨a,b|a4,b2,(a⁢b)2⟩ – the dihedral groupMathworldPlanetmath of order 8, and Q8=⟨i,j|i4,i2=j2,ij=i-1⟩ – the quaternion groupMathworldPlanetmathPlanetmath of order 8, then D8∘D8=D8×D8/⟨(a2,a2)⟩ is isomorphic to Q8∘Q8=(Q8×Q8)/⟨(i2,i2)⟩; yet, D8 and Q8 are nonisomorphic and centrally indecomposable. In particular, central decompositions are not unique even up to automorphismsPlanetmathPlanetmathPlanetmathPlanetmath. This is in contrast the well-known Krull-Remak-Schmidt theorem for direct products of groups.

3 History

The name central product appears to have been coined by Philip Hall [1, SectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 3.2] though the principal concept of such a productMathworldPlanetmath appears in earlier work (e.g. [2, Theorem II]). Hall describes central products as “…the group obtained from the direct product by identifying the centres of the direct factors…”. The modern definition clearly out grows this original version as now centers may be only partially identified.

References

  • 1 P. Hall, Finite-by-nilpotent groups, Proc. Camb. Phil. Soc., 52 (1956), 611-616.
  • 2 B. H. Neumann, and H. Neumann, A remark on generalized free productsMathworldPlanetmath, J. London Math. Soc. 25 (1950), 202-204.
Title central product of groups
Canonical name CentralProductOfGroups
Date of creation 2013-03-22 18:49:45
Last modified on 2013-03-22 18:49:45
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 6
Author Algeboy (12884)
Entry type Definition
Classification msc 20E34
Synonym central decomposition
Defines central decomposition
Defines central product