Covering the alternating groups by products of cycle classes |
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Authors: | Marcel Herzog |
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Affiliation: | a School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, Israel b School of Computer Sciences, The Academic College of Tel-Aviv-Yafo, 2 Rabenu Yeruham St., Tel-Aviv 61083, Israel |
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Abstract: | Given integers k,l?2, where either l is odd or k is even, we denote by n=n(k,l) the largest integer such that each element of An is a product of k cycles of length l. For an odd l, k is the diameter of the undirected Cayley graph Cay(An,Cl), where Cl is the set of all l-cycles in An. We prove that if k?2 and l?9 is odd and divisible by 3, then . This extends earlier results by Bertram [E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory 12 (1972) 368-380] and Bertram and Herzog [E. Bertram, M. Herzog, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (2001) 87-99]. |
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Keywords: | Alternating groups Products of cycles Covering number |
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