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

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.

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
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