OUTER AUTOMORPHISM GROUPS OF FINITE -NILPOTENT GROUPS

ABSTRACT

When does a finite group possess a non-trivial outer automorphism? In this paper for a finite -nilpotent group we shall prove the following results.

a.	If , then divides the order of
b.	If is non-abelian, then divides the order of .

Our results extend a famous theorem of Gaschütz and can be applied to the investigation of finite complete groups.

ACKNOWLEDGMENTS

The author wishes to express his gratitude to Professor Xiuyun Guo for many helpful and stimulating conversations during the course of this work. Also, the author is very grateful to Professor Derek J.S. Robinson for good suggestions about the modification of the paper and for giving the simplified proofs of Lemma 4.1 and Theorem A in this paper.