On the Exterior Degree of Finite Groups

Abstract

We introduce the exterior degree of a finite group G to be the probability for two elements g and g′ in G such that g ∧ g′ = 1, and we shall state some results concerning this concept. We show that if G is a non-abelian capable group, then its exterior degree is less than 1/p, where p is the smallest prime number dividing the order of G. Finally, we give some relations between the new concept and commutativity degree, capability, and the Schur multiplier.

Key Words:

• Capable group
• Commutativity degree
• Exterior centre
• Exterior degree
• Schur multiplier

2000 Mathematics Subject Classification:

• 20F99
• 20P05

ACKNOWLEDGMENTS

This work was started during the visit of authors to the Department of Mathematics, University of Naples Federico II. We would like to thank Professor Francesco De Giovanni for his comments and encouragement.

Notes

Communicated by P. Tiep.