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Solubilizers in profinite groups.
- Source :
-
Journal of Algebra . Jun2024, Vol. 647, p619-632. 14p. - Publication Year :
- 2024
-
Abstract
- The solubilizer of an element x of a profinite group G is the set of the elements y of G such that the subgroup of G generated by x and y is prosoluble. We propose the following conjecture: the solubilizer of x in G has positive Haar measure if and only if x centralizes 'almost all' the non-abelian chief factors of G. We reduce the proof of this conjecture to another conjecture concerning finite almost simple groups: there exists a positive c such that, for every finite simple group S and every (a , b) ∈ (Aut (S) ∖ { 1 }) × Aut (S) , the number of s is S such that 〈 a , b s 〉 is insoluble is at least c | S |. Work in progress by Fulman, Garzoni and Guralnick is leading to prove the conjecture when S is a simple group of Lie type. In this paper we prove the conjecture for alternating groups. [ABSTRACT FROM AUTHOR]
- Subjects :
- *FINITE simple groups
*PROFINITE groups
*HAAR integral
*NONABELIAN groups
Subjects
Details
- Language :
- English
- ISSN :
- 00218693
- Volume :
- 647
- Database :
- Academic Search Index
- Journal :
- Journal of Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 176296814
- Full Text :
- https://doi.org/10.1016/j.jalgebra.2024.01.048