On the Order of the Automorphism Group of a Finite Group

Abstract

We show that if |Aut(G)| is odd, then without loss of generality we may assume the only primes dividing |G| are those dividing |Aut(G)|. We also classify finite abelian groups A such that |Aut(A)| = |A| and finite abelian groups A such that |Aut(A)| _p = |A| _p , where p is a prime dividing |A|.

Key Words:

• Automorphism
• Finite Group

2000 Mathematics Subject Classification:

• Primary 20F28, 20D45
• Secondary 20K30

ACKNOWLEDGMENT

The second author gratefully acknowledges the support provided by the King Fahd University of Petroleum and Minerals during this research.

Notes

Communicated by J. Zhang.