central product of groups
for , if, and only if, , and
for all .
A group is centrally indecomposable if its only central decomposition is . A central decomposition is fully refined if its members are centrally indecomposable.
Condition 1 is often relaxed to but this has the negative affect of allowing, for example, to have the central decomposition such sets as and in general a decomposition of any possible size. By impossing 1, we then restrict the central decompositions of to direct decompositions. Furthermore, with condition 1, the meaning of indecomposable is easily had.
A central product is a group where is a normal subgroup of and for all .
Every finite central decomposition is a central product of the members in .
Suppose that is a a finite central decomposition of . Then define by . Then . Furthermore, for each direct factor of . Thus, is a central product of . ∎
Every direct product is also a central product and so also every direct decomposition is a central decomposition. The converse is generally false.
Let , for a field . Then is a centrally indecomposable group.
then is a central decomposition of . Furthermore, each is isomorphic to and so is a fully refined central decomposition.
If – the dihedral group of order 8, and – the quaternion group of order , then is isomorphic to ; yet, and are nonisomorphic and centrally indecomposable. In particular, central decompositions are not unique even up to automorphisms. This is in contrast the well-known Krull-Remak-Schmidt theorem for direct products of groups.
The name central product appears to have been coined by Philip Hall [1, Section 3.2] though the principal concept of such a product 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.
- 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 products, J. London Math. Soc. 25 (1950), 202-204.
|Title||central product of groups|
|Date of creation||2013-03-22 18:49:45|
|Last modified on||2013-03-22 18:49:45|
|Last modified by||Algeboy (12884)|