The Tarski irredundant basis theorem and the finite soluble groups. Issue 5 (17th December 2018)

• Record Type: Journal Article
• Authors: Lucchini, Andrea; Moscatiello, Mariapia
• Is Part Of: Mathematische Nachrichten. Volume 292:Issue 5(2019)
• Page Start: 1022
• Page End: 1031
• Publication Date: 2018-12-17
• Subjects: generating sets -- soluble groups -- Tarski theorem -- 20F05; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201800346
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: Denote by d = d ( G ) and m = m ( G ), respectively, the smallest and the largest cardinality of a minimal generating set of a finite group G . The Tarski irredundant basis theorem implies that for every k with d ( G ) ≤ k < m ( G ) there exist a minimal generating set ω = { g 1, ⋯, g k }, an index i ∈ { 1, ⋯, k } and x 1, x 2 in G such that ω ∼ = { g 1, ⋯, g i − 1, x 1, x 2, g i + 1, ⋯, g k } is again a minimal generating set of G . In this case we say that ω ∼ is an immediate descendant of ω. There are several examples of minimal generating sets of cardinality smaller than m ( G ) which have no immediate descendant and so it appears an interesting problem to investigate under which conditions an immediate descendant exists. In this paper we discuss this problem in the case of finite soluble groups.