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Rewriting products of group elements. II. (English) Zbl 0663.20036
A group has property \(Q_ n\) (is n-rewritable) if for every choice \(x_ 1,...,x_ n\) of elements in G there exist distinct permutations \(\sigma\) and \(\tau\) on n letters such that \(x_{\sigma (1)}...x_{\sigma (n)}=x_{\tau (1)}...x_{\tau (n)}\). Theorem 2 is the main result of the paper. It states that 4-rewritable groups are soluble. Using the fact that n-rewritable groups are finite-by-abelian-by-finite [see the author in part I, J. Algebra 116, No.2, 506-521 (1988; Zbl 0647.20033)], the proof of this is reduced to checking the minimal simple groups. For some small cases, CAYLEY was used.
In Paragraph 2, it is shown that an n-rewritable semisimple group is finite of bounded order (depending solely on n). The argument depends on the CLASSIFICATION, however, an elementary proof of this fact meanwhile is available, see p. 81 of the article reviewed below.
Reviewer: R.Brandl

MSC:
20F12 Commutator calculus
20F16 Solvable groups, supersolvable groups
20F24 FC-groups and their generalizations
20F05 Generators, relations, and presentations of groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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