Vol. 18, No. 1, 1966

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The Klein group as an automorphism group without fixed point

Steven Fredrick Bauman

Vol. 18 (1966), No. 1, 9–13

An automorphism group V acting on a group G is said to be without fixed points if for any g G, v(g) = g for all v V implies that g = 1. The structure of V in this case has been shown to influence the structure of G. For example if V is cyclic of order p and G finite then John Thompson has shown that G must be nilpotent. Gorenstein and Herstein have shown that if V is cyclic of order 4 then a finite group G must be solvable of p-length 1 for all p||G| and G must possess a nilpotent commutator subgroup.

In this paper we will consider the case where G is finite and V noncyclic of order 4. Since V is a two group all the orbits of G under V save the identity have order a positive power of 2. Thus G is of odd order and by the work of Feit-Thompson G is solvable. We will show that G has p-lengh 1 for all p||G| and G must possess a nilpotent commutator subgroup.

Mathematical Subject Classification
Primary: 20.20
Secondary: 20.22
Received: 3 March 1965
Published: 1 July 1966
Steven Fredrick Bauman