Fixed Point Free Action with Some Regular Orbits

✍ Scribed by Alexandre Turull

• Book ID: 102574210
• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 247 KB
• Volume: 194
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7035

✦ Synopsis

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.

Ž< < < <. Assume furthermore that A , G s 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 w Ž .

x A. In A. Turull, J. Reine Angew. Math. 371 1986 , 67᎐91 , this was proved in the case where for every proper subgroup B of A and every B-invariant elementary Ž . Ž . Ž abelian section S of G, there exists some ¨g S such that C ¨s C 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.