On Minimal Orders of Groups with Odd Order Automorphism Groups

Abstract

Let G be a finite group. Suppose that the number of prime factors of the order of G, counting repetitions, is less than or equal to five and that the order of G is not equal to p ⁵ for some prime number p > 3. Then we show that the order of the automorphism group of G is even. We finish with some conjectures on the smallest groups with certain odd order automorphism groups.

Key Words:

• Automorphism group
• N.I. group

2000 Mathematics Subject Classification:

• Primary 20D45
• Secondary 20D15

ACKNOWLEDGMENT

The second author would like to thank the staff of the Department of Mathematics, University of Otago, New Zealand for the support received during his sabbatical stay in 2009.

Notes

Communicated by J. Zhang