Fixed point free action with some regular orbits

[摘要] Let the finite group A be acting on a finite (solvable) group G and suppose that no non-trivial element of G is fixed under the action of all the elements of A. Assume furthermore that (\A\, \G\) = 1. A long-standing conjecture is that then the Fitting height of G is bounded by the length of the longest chain of subgroups of A. In [A. Turull, J. Reine Angew. Math. 371 (1956), 67-91], this was proved in the case where for every proper subgroup B of A and every B-invariant elementary abelian section S of G, there exists some upsilon is an element of S such that C-B(upsilon) = C-B(S) (we say that B has a regular orbit on S). In the present paper we establish the conjecture assuming only that some of these sections have regular orbits. (C) 1997 Academic Press.

[发布日期] 1997-08-15