Author:
Burness Timothy C.,Guralnick Robert M.
Abstract
AbstractLet G be a finite group, let H be a core-free subgroup and let b(G, H) denote the base size for the action of G on G/H. Let $$\alpha (G)$$
α
(
G
)
be the number of conjugacy classes of core-free subgroups H of G with $$b(G,H) \ge 3$$
b
(
G
,
H
)
≥
3
. We say that G is a strongly base-two group if $$\alpha (G) \le 1$$
α
(
G
)
≤
1
, which means that almost every faithful transitive permutation representation of G has base size 2. In this paper, we study the strongly base-two finite groups with trivial Frattini subgroup.
Funder
National Science Foundation
Simons Foundation
Publisher
Springer Science and Business Media LLC
Reference34 articles.
1. Aivazidis, S., Ballester-Bolinches, A.: On the Frattini subgroup of a finite group. J. Algebra 470, 254–262 (2017)
2. Aschbacher, M.: Finite Group Theory. Cambridge Studies in Advanced Mathematics, vol. 10. Cambridge University Press, Cambridge (1986)
3. Bailey, R.F., Cameron, P.J.: Base size, metric dimension and other invariants of groups and graphs. Bull. Lond. Math. Soc. 43, 209–242 (2011)
4. Bennett, M.A., Skinner, C.M.: Ternary Diophantine equations via Galois representations and modular forms. Can. J. Math. 56, 23–54 (2004)
5. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symb. Comput. 24, 235–265 (1997)