INSOLUBLE GROUPS WITH THE REWRITING PROPERTY P8

[摘要] Let n be an integer greater than 1, and let G be a group. An n-tuple x1, x2, ..., x(n) of elements of G is called rewritable if there is a nontrivial permutation-pi of {1, 2, ..., n} such that x1x2...x(n) = x-pi(1)x-pi(2)...x-pi(n). The group G is said to have the rewriting property P(n) if every n-tuple of G is rewritable. The authors have previously shown that all groups with P7 are soluble, whereas the alternating group A5 has P8. Thus the least n for which P(n) contains a nonabelian simple group is 8. In this article the authors confirm conjectures of R. Brandl concerning the structure of insoluble groups with P8. In particular, let G be an insoluble group. Then G has the rewriting property P8 if and only if it is a semidirect product H times sign with bar connected to left K where H is abelian, K is isomorphic with the alternating group of degree 5, and \H:C(H)(K)\ = 1 or 2.

[发布日期] 1991-07-26