Finite groups of bounded rank with an almost regular automorphism of prime order. (Russian, English) Zbl 1018.20017

Khukhro, E. I.

Sib. Mat. Zh. 43, No. 5, 1182-1191 (2002); translation in Sib. Math. J. 43, No. 5, 955-962 (2002).

The main result reads as follows: Let \(G\) be a finite group of rank \(r\), let \(\varphi\) be an automorphism, and let \(|C_G(\varphi)|=m\). Then there is a \(\varphi\)-invariant nilpotent subgroup \(H\) and the index \(G:H\) is an \((r,m)\)-bounded number and the nilpotency class of \(H\) is an \(r\)-bounded number. If \(m=1\) then the group \(G\) is a nilpotent group of \(r\)-bounded nilpotency class.

Reviewer: K.N.Ponomarev (Novosibirsk)

MSC:

20D45	Automorphisms of abstract finite groups
20D15	Finite nilpotent groups, \(p\)-groups
20D60	Arithmetic and combinatorial problems involving abstract finite groups
20F40	Associated Lie structures for groups

Keywords:

regular automorphisms; rank of finite groups; nilpotent subgroups; nilpotency classes; associated Lie rings