Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group

Abstract

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.

Key Words:

• Finite simple groups
• Automorphism groups

Mathematics Subject Classification:

• 20E32
• 20F28
• 20-04

Acknowledgments

The author would like to thank Wolfgang Kimmerle and Frank Lübeck for their valuable support, useful hints and many interesting discussions.

Notes

^#Communicated by H.-J. Schneider.