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