Fixed point free action on groups of odd order

Abstract

Let A be a finite abelian group that acts fixed point freely on a finite (solvable) group G . Assume that |G| is odd and A is of squarefree exponent coprime to 6. We show that the Fitting length of G is bounded by the length of the longest chain of subgroups of A

Fixed-point free action of an abelian group of odd non-squarefree exponent

“…However, by employing the ideas of both [3] and [9], we have been able to obtain Theorem A in [5]. In the present paper, we take this result further by slightly modifying the techniques used in [5]; namely, we obtain the following theorem.…”

“…If A is nilpotent, it is expected that the conjecture is still true without the coprimeness condition. This question is settled for cyclic groups A of order pq [2] and pqr [4] for pairwise distinct primes p, q and r. We also establish the conjecture without the coprimeness condition in the case where A is abelian of squarefree exponent coprime to 6 and |G| is odd [5]. The main theorems of [2] and [4] have the advantage of providing an answer in the absence of any restriction on the nature of the primes involved.…”

“…When |A| is divisible by the primes 2 or 3, the study of such problems needs much more effort. Our Theorem C allows |A| to be odd and divisible by 3, in contrast to the assumption in [5] that |A| is coprime to 6.…”

“…Theorem A is a generalization of Theorem A in [5] that replaces the hypothesis that |G| is odd by the weaker one that the Sylow 2-subgroups of G are abelian. While this is a theorem that was designated as a partial special case in the survey article [10] by Turull, Theorem B is a full special case theorem in the sense of the same article.…”

The Fitting length of finite soluble groups II

“…If W = 3, then |ϕ| = pqr (p, q, r primes, p = q = r = p) and h(G) ≤ 3 follows from [5]. We want to recall that a result of Ercan and Güloglu (Theorem A of [6]) asserts that if G has odd order, A is abelian of squarefree exponent coprime to 6 and C G (A) = 1, then h(G) ≤ W (A).…”