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Groups with a nilpotent-by-finite triple factorization. (English) Zbl 0629.20016

In many papers on factorized groups one has to study groups \(G=HK=HA=KA\) which have a triple factorization as a product of two subgroups H and K and a normal subgroup A of G. Therefore it is of particular interest to know whether G satisfies some nilpotency requirement whenenver the three subgroups H, K and A satisfy this same nilpotency requirement. A positive answer to this problem is given for the classes of finite-by-nilpotent and nilpotent-by-finite groups provided that G satisfies certain finiteness conditions. The proofs of this paper make use of cohomology theory.

MSC:

20F19 Generalizations of solvable and nilpotent groups
20E22 Extensions, wreath products, and other compositions of groups
20D40 Products of subgroups of abstract finite groups
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