## 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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