# Finite p -groups with a Frobenius group of automorphisms whose kernel is a cyclic p -group

@inproceedings{Khukhro2013FiniteP, title={Finite p -groups with a Frobenius group of automorphisms whose kernel is a cyclic p -group}, author={E. I. Khukhro and N. Yu. Makarenko}, year={2013} }

Suppose that a finite $p$-group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ that is a cyclic $p$-group and with complement $H$. It is proved that if the fixed-point subgroup $C_G(H)$ of the complement
is nilpotent of class $c$, then $G$ has a characteristic subgroup of index bounded in terms of $c$,
$|C_G(F)|$, and $|F|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. Examples show that the condition of $F$ being cyclic is essential.
The proof is based…

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## 32 References

### Structure and derived length of finite $p$-groups possessing an automorphism of $p$-power order having exactly $p$ fixpoints.

- Mathematics
- 1988

Everywhere in this paper p denotes a prime number. In [1] Alperin showed that the derived length of a finite p-group possessing an automorphism of order p and having exactly p fixed points is bounded…

### FINITE $ p$-GROUPS ADMITTING $ p$-AUTOMORPHISMS WITH FEW FIXED POINTS

- Mathematics
- 1995

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- Mathematics
- 2013

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- Mathematics
- 2013

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- Mathematics
- 2010

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- Mathematics
- 2011

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### Automorphisms of solvable groups

- Mathematics
- 1975

We state an analogue of Tits' theorem on linear groups [3] as CONJECTURE. Let G be a finitely generated (f.g.) solvable group. Then, any f.g. subgroup of the automorphism group of G is solvable by…

### Lie algebras admitting a metacyclic frobenius group of automorphisms

- Mathematics
- 2013

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### On Almost Regular Automorphisms of Finite p-Groups

- Mathematics
- 2000

Abstract In this paper we prove that there are functions f ( p , m , n ) and h ( m ) such that any finite p -group with an automorphism of order p n , whose centralizer has p m points, has a…