Year: 2026
Type: article
Abstract: Famously, every finite simple group can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate tends to 1 as . In this paper, we generalize this theorem of Liebeck and Shalev. Work of Lucchini and Menegazzo (1997) implies that a finite group can be generated by a pair of elements if it has a unique chief series. As a consequence of our main theorem, the probability that a pair of elements generate such a group tends to 1 as , where is the unique simple quotient of . We also prove that a profinite group with finitely many chief series has a topological generating set of size , and for any such , the probability that a ‐tuple of elements topologically generates is positive; moreover, we can take if has a unique chief series. Along the way, we show that the chief factors of a finite group with a unique chief series are highly constrained, and we also analyze the maximal subgroup zeta function of a finite group with a unique minimal normal subgroup.
DOI: 10.1112/jlms.70670
Authors: Scott Harper, Martyn Quick
Institutions: University of Birmingham, University of St Andrews
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,930
OpenAlex ID: W7125760313
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