zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] Bianchi, M; Brandi, R; Mauri, A.G.B, On the 4-permutational property for groups, Arch. math. (basel), 48, 281-285, (1987) · Zbl 0623.20022
[2] Blyth, R.D, Rewriting products of group elements—I, J. algebra, 116, 506-521, (1988) · Zbl 0647.20033
[3] Bourbaki, N, Élements de mathématique, (), Fasc. XXXIV · Zbl 0165.56403
[4] Carter, R.W, Simple groups of Lie type, () · Zbl 0232.20017
[5] Chevalley, C, Sur certains groupes simples, Tôhoku math. J. (2), 7, 14-66, (1955) · Zbl 0066.01503
[6] Huppert, B; Blackburn, N, Finite groups III, () · Zbl 0514.20002
[7] Jones, G.A, Varieties and simple groups, J. austral. math. soc., 17, 163-173, (1974) · Zbl 0286.20028
[8] {\scP. Longobardi and M. Maj}, Sui gruppi per cui ogni prodotto di quattro elementi è riordinabile, preprint.
[9] Lüneburg, H, Some remarks concerning the ree groups of type (G2), J. algebra, 3, 256-259, (1966) · Zbl 0135.39401
[10] Robinson, D.J.S, A course in the theory of groups, () · Zbl 0496.20038
[11] Suzuki, M, Group theory I, ()
[12] Thompson, J.G, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. amer. math. soc., 74, 383-437, (1968) · Zbl 0159.30804
[13] Tits, J, LES groupes simples de Suzuki et de ree, (), exposé 210 · Zbl 0267.20041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.