Groups with regular elementary 2-groups of automorphisms

Abstract

In the early 1960's there arose in the theory of finite groups the following Conjecture. Suppose ~ is a finite solvable group, V is a subgroup of A~t~ C~(V) = ~ (IVI~I~I)=4 , and ]Vl is the product of /% primes, not necessarily distinct. Then the nilpotent length of ~ is at most /%.It is well known that if the pair V , ~ satisfies the conditions of the conjecture and ]Vl--2, then ~ is Abelian. Shult [4] showed that the conjecture is true if V is an elementary 2-group. Bauman [2] proved that if V is a four-group, then the commutant of 0 is nilpotent. Many other cases of the problem have also been studied [3,5,6]. Almost all of the results depend on the theory of representations of finite groups. In the present paper we suggest another approach, which does not require that G be finite. Here consider the case where V is an elementary 2-group and ~ is periodic. It is easy to see that these results are stronger than those of Shult and Bauman. Moreover, they show that in some cases the conjecture can be significantly strengthened.In connection with Theorem 1 it is appropriate to mention that for any integers fL >i 2 and ~< >.~ there exists a K-step solvable, periodic group admitting a regular elementary group of automorphisms of order ~a.We also mention that the approach suggested in this paper enables us to obtain a generalization of the theorem of Kreknin and Kostrikin [i] which says that the nilpotent length of a K-step solvable Lie algebra admitting a regular automorphism of prime order P does not exceed some number A(~ K} depending only on p and K . It can be shown that a #<-step solv-

Applications of Lie ring methods to group theory

“…Thus, a group is 1-finite if and only if it is periodic. It was proved in [37] that the equality C G/N (A) = C G (A)N/N holds whenever A is a 2-group. More generally, we have.…”

On the exponent of a finite group admitting a fixed-point-free four-group of automorphisms

“…Put G i = C G (v i ) for i ∈ {1, 2, 3}. The proofs of the next few lemmas can be found in [15]. Lemma 4.1.…”

On Groups of Odd Order Admitting an Elementary 2-Group of Automorphisms

“…It is well-known that very often the structure of C G (A) has strong influence over the structure of the whole group G. The influence seems especially strong in the case where G is a finite group of odd order and A is an elementary abelian 2-group. It was shown in [5] that if G is a finite group of derived length k on which an elementary abelian group A of order 2 n acts fixedpoint-freely, then G has a normal series G N 0 ! Á Á Á !…”