The normal covering number γ(G) of a finite, non-cyclic group G is the least number of proper subgroups such that each element of G lies in some conjugate of one of these subgroups. We prove that there is a positive constant c such that, for G a symmetric group Sym(. n) or an alternating group Alt(. n), γ(. G) ≥ c n. This improves results of the first two authors who had earlier proved that aφ(. n) ≤ γ(. G) ≤ 2. n/3, for some positive constant a, where φ is the Euler totient function. Bounds are also obtained for the maximum size κ(. G) of a set X of conjugacy classes of G = Sym(. n) or Alt(. n) such that any pair of elements from distinct classes in X generates G, namely c n ≤ κ(. G) ≤ 2. n/3. © 2013 Elsevier Inc.

Bubboloni, D., Praeger, C., Spiga, P. (2013). Normal coverings and pairwise generation of finite alternating and symmetric groups. JOURNAL OF ALGEBRA, 390, 199-215 [10.1016/j.jalgebra.2013.05.017].

Normal coverings and pairwise generation of finite alternating and symmetric groups

SPIGA, PABLO
2013

Abstract

The normal covering number γ(G) of a finite, non-cyclic group G is the least number of proper subgroups such that each element of G lies in some conjugate of one of these subgroups. We prove that there is a positive constant c such that, for G a symmetric group Sym(. n) or an alternating group Alt(. n), γ(. G) ≥ c n. This improves results of the first two authors who had earlier proved that aφ(. n) ≤ γ(. G) ≤ 2. n/3, for some positive constant a, where φ is the Euler totient function. Bounds are also obtained for the maximum size κ(. G) of a set X of conjugacy classes of G = Sym(. n) or Alt(. n) such that any pair of elements from distinct classes in X generates G, namely c n ≤ κ(. G) ≤ 2. n/3. © 2013 Elsevier Inc.
Articolo in rivista - Articolo scientifico
Conjugacy classes; Generation of groups; Normal coverings; Symmetric groups;
English
2013
390
199
215
none
Bubboloni, D., Praeger, C., Spiga, P. (2013). Normal coverings and pairwise generation of finite alternating and symmetric groups. JOURNAL OF ALGEBRA, 390, 199-215 [10.1016/j.jalgebra.2013.05.017].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/48990
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