Covering the alternating groups by products of cycle classes.
type: Item
instance of: scholarly article
title: Covering the alternating groups by products of cycle classes.
zbMATH Open document ID: 1172.20002
author: Marcel Herzog
author: Gil Kaplan
author: Arieh Lev
published in: Journal of Combinatorial Theory. Series A
publication date: 2008-12-08T00:00:00Z
review text: Let \(A_n\) denote the group of even permutations on \(n\) letters. Given integers \(k,l\geq 2\) the integer \(n=n(k,l)\) is defined as the largest integer such that every permutation in \(A_n\) is a product of \(k\) cycles of length \(l\). The authors prove several interesting generalizations of earlier results due to \textit{E. Bertram} [J. Comb. Theory, Ser. A 12, 368-380 (1972; Zbl 0238.20004)] and \textit{E. Bertram} and \textit{M. Herzog} [J. Comb. Theory, Ser. A 94, No. 1, 87-99 (2001; Zbl 0983.20003)]. E.g. the following main result is proved. Theorem 3.4. Let \(k\) and \(l\) be such that \(k\geq 2\) and \(l\geq 9\) is odd and divisible by \(3\). Then \({2\over 3}kl\leq n(k,l)\leq {2\over 3}kl+1\). Furthermore, if \(k\) is odd then \(n(k,l)={2\over 3}kl\). Theorem 3.3 contains a result for arbitrary \(l>2\) which is more complicated in detail. Of course, the results are closely related to the covering numbers of conjugacy classes in \(A_n\).
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 5379250
zbMATH Keywords: alternating groups
zbMATH Keywords: conjugacy classes
zbMATH Keywords: products of cycles
zbMATH Keywords: symmetric groups
zbMATH Keywords: covering numbers
MaRDI profile type: Publication
full work available at URL: j.jcta.2008.01.010
OpenAlex ID: W1988998671
cites work: Q3695451
cites work: Even permutations as a product of two conjugate cycles
cites work: Powers of cycle-classes in symmetric groups
cites work: Q3475451
cites work: Products of conjugacy classes in groups
cites work: Representation of permutations as products of two cycles
cites work: On the diameter of a Cayley graph of a simple group of Lie type based on a conjugacy class
cites work: Q5480921
cites work: Q5331572
cites work: Mixing and covering in the symmetric groups
DOI: 10.1016/J.JCTA.2008.01.010
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description: scientific article; zbMATH DE number 5379250