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Triply factorized groups. (English) Zbl 0732.20014

Groups, Vol. 1, Proc. Int. Conf., St. Andrews/UK 1989, Lond. Math. Soc. Lect. Note Ser. 159, 1-13 (1991).
[For the entire collection see Zbl 0722.00007.]
If the group \(G=AB\) is the product of two subgroups A and B, and N is a normal subgroup of G, the factorizer \(X(N)=AN\cap BN\) of N has the triple factorization \(X(N)=(A\cap BN)(B\cap AN)=(A\cap BN)N=(B\cap AN)N.\) Therefore, in order to get information on the structure of a factorized group it is often useful to consider groups having a triple factorization of the form \(G=AB=AK=BK\), where K is normal in G. This survey article especially deals with the structure of groups triply factorized by (generalized) nilpotent or supersoluble subgroups. Some theorems on this problem are reported, which are mainly due to the author, S. Franciosi and the reviewer. Moreover, a relevant example by Ya. Sysak [Akad. Nauk Ukr. Inst. Mat., Kiev, Preprint 82.53 (1982)] of a non-(locally nilpotent) group factorized by three abelian subgroups is constructed.

MSC:

20E22 Extensions, wreath products, and other compositions of groups
20F18 Nilpotent groups
20D40 Products of subgroups of abstract finite groups

Citations:

Zbl 0722.00007