# FINITE GROUPS ADMITTING A FIXED-POINT-FREE AUTOMORPHISM OF ORDER 4.*

@article{Gorenstein1961FINITEGA, title={FINITE GROUPS ADMITTING A FIXED-POINT-FREE AUTOMORPHISM OF ORDER 4.*}, author={Daniel Gorenstein and Israel N. Herstein}, journal={American Journal of Mathematics}, year={1961}, volume={83}, pages={71} }

Recently, in a remarkable piece of work [4, 5] John Thompson has proved a result which implies as an immediate corollary the well-known Frobenius conjecture, namely that a finite group admitting a fixed-point-free automorphism (i. e., leaving only the identity element fixed) of prime order must be nilpotent. However, non-nilpotent groups are known which admit fixedpoint-free automorphisms of composite order. In all these cases one notices that the groups in question are solvable. Although the…

## 45 Citations

### Nilpotent height of finite groups admitting fixed-point-free automorphisms

- Mathematics
- 1964

w 1. Introduction In this paper, "group" is to mean finite group. We shall consider certain properties of groups admitting fixed-point-free automorphisms (that is, automorphisms which leave only the…

### Automorphisms of Finite Groups and their Fixed-Point Groups

- MathematicsJournal of the Australian Mathematical Society
- 1969

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows…

### Some Remarks on Commuting Fixed Point Free Automorphisms of Groups

- Mathematics
- 2016

Introduction Let be a group. By and we denote the ce nter, the commutator subgroup, the group of all automorphisms, the group of central aut omorphisms, the group of all inner automorphisms and the…

### SOLVABLE GROUPS ADMITTING A FIXED-POINT-FREE AUTOMORPHISM OF PRIME POWER ORDER

- Mathematics
- 1966

Here h(G), the Fitting height (also called the nilpotent length) of G, is as defined in [7]. 1I(G), the r-length of G, is defined in an obvious analogy to the definition of p-length in [2]. Higman…

### Nilpotence of the commutator subgroup in groups admitting fixed point free operator groups.

- Mathematics
- 1966

Let V be a group of operators acting in fixed point free manner on a group G and suppose V has order relatively prime to I G I. Work of several authors has shown that if V is cyclic of prime order or…

### The Klein group as an automorphism group without fixed point.

- Mathematics
- 1966

An automorphism group V acting on a group G is said to be without fixed points if for any g eG, v(g) = g for all v eV implies that g = 1. The structure of V in this case has been shown to influence…

### The classification of finite simple groups I. Simple groups and local analysis

- Mathematics
- 1979

It is indeed unfortunate that Richard Brauer did not live to see the complete classification of the finite simple groups. He had devoted the past thirty years largely to their study and it is…

### The nilpotency of finite groups with an automorphism satisfying an identity

- Mathematics
- 2018

We generalise the positive solution of the Frobenius conjecture (by J. Thompson) and refinements thereof (by Higman, Kreknin, and Kostrikin). This allows us to also extend the positive solution of…

## References

SHOWING 1-3 OF 3 REFERENCES

### Finite Groups Which Admit An Automorphism With Few Orbits

- MathematicsCanadian Journal of Mathematics
- 1960

In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an…

### FINITE GROUPS WITH FIXED-POINT-FREE AUTOMORPHISMS OF PRIME ORDER.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1959