Extendible elements of the alternating groups

Abstract

An element σ of A_n , the Alternating group of degree n, is extendible in S_n , the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.

Keywords:

• Alternating group
• Extendible element
• Cycle-decomposition

Keywords:

• AMS 2000 Mathematics Subject Classifications:
• 20B30; 20B35

Acknowledgments

This work was undertaken within the “Centro de Estruturas Lineares e Combinatáorias” of the University of Lisbon. The second author was partially supported by the Gulbenkian Foundation. The authors thank Dr Chris Parker for suggesting a simplification of the original proof of Lemma 7 and the referees for their very helpful suggestions.