2013/11/24 by Cristina Acciarri, Acciarri, Cristina, Pavel Shumyatsky +1 · 1 citation
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary 20F14 #Secondary 20D25 #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1311.6148
openalex publication_date 2013/11/24 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
The coprime commutators \γj^* and \δj^* were recently\nintroduced as a tool to study properties of finite groups that can be expressed\nin terms of commutators of elements of coprime orders. They are defined as\nfollows. Let G be a finite group. Every element of G is both a\n\γ1^*-commutator and a \δ0^*-commutator. Now let j\≥ 2 and\nlet X be the set of all elements of G that are powers of\n\γj-1^*-commutators. An element g is a \γj^*-commutator if\nthere exist a\∈ X and b\∈ G such that g=[a,b] and (|a|,|b|)=1. For\nj\≥ 1 let Y be the set of all elements of G that are powers of\n\δj-1^*-commutators. The element g is a \δj^*-commutator if\nthere exist a,b\∈ Y such that g=[a,b] and (|a|,|b|)=1. The subgroups of\nG generated by all \γj^*-commutators and all \δj^*-commutators\nare denoted by \γj^*(G) and \δj^*(G), respectively. For every\nj\≥2 the subgroup \γj^*(G) is precisely the last term of the lower\ncentral series of G (which throughout the paper is denoted by\n\γ_\∞(G)) while for every j\≥1 the subgroup \δj^*(G) is\nprecisely the last term of the lower central series of \δj-1^*(G),\nthat is, \δj^*(G)=\γ_\∞(\δj-1^*(G)).\n In the present paper we prove that if G possesses m cyclic subgroups\nwhose union contains all \γj^*-commutators of G, then \γj^*(G)\ncontains a subgroup \Δ, of m-bounded order, which is normal in G and\nhas the property that \γj*(G)/\Δ is cyclic. If j\≥2 and G\npossesses m cyclic subgroups whose union contains all\n\δj^*-commutators of G, then the order of \δj^*(G) is\nm-bounded.\n