TY - JOUR

T1 - Covering the alternating groups by products of cycle classes

AU - Herzog, Marcel

AU - Kaplan, Gil

AU - Lev, Arieh

PY - 2008/10

Y1 - 2008/10

N2 - 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 frac(2, 3) k l ≤ n (k, l) ≤ frac(2, 3) k l + 1. 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].

AB - 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 frac(2, 3) k l ≤ n (k, l) ≤ frac(2, 3) k l + 1. 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].

KW - Alternating groups

KW - Covering number

KW - Products of cycles

UR - http://www.scopus.com/inward/record.url?scp=50649088926&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2008.01.010

DO - 10.1016/j.jcta.2008.01.010

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AN - SCOPUS:50649088926

SN - 0097-3165

VL - 115

SP - 1235

EP - 1245

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

IS - 7

ER -