1991 Mathematics Subject Classification: Primary 20F17,20D40; Secondary 20D25, 20E28.
For not explicitly explained concepts used in this report, the cited literature or usual books in finite group theory may be consulted.
Based on W. Burnside's famous theorem, published in 1904 [5], which states that a finite group is soluble, if its order is divisible by at most two distinct prime numbers, P. Hall characterizes the soluble groups in the decade 1928 – 1937, by means of the existence of Sylow complements and Sylow systems [15, 16, 17]. In particular:
A finite group is soluble if and only if it is the product of pairwise permutable Sylow subgroups.
With this characterization arises the question:
If a finite group is the product of pairwise permutable nilpotent subgroups, the group is soluble?
In 1951, H. Wielandt [28] begins with a series of theorems which he finishes in 1958 [29] and which together with a result of O. H. Kegel 1962 [21] answers affirmatively to this question.
Meanwhile, in 1953, B. Huppert [19] solves a particular case of the problem, which merits to be mentioned:
A finite group which is the product of pairwise permutable cyclic subgroups is supersoluble.