Transitive Subgroups of Primitive Permutation Groups
By Martin W. Liebeck, Cheryl E. Praeger, Jan Saxl. 2000
99 citations recorded by the source corpus.
Abstract We investigate the finite primitive permutation groups G which have a transitive subgroup containing no nontrivial subnormal subgroup of G . The conclusion is that such primitive groups are rather rare, and that their existence is intimately connected with factorisations of almost simple groups. A corollary is obtained on primitive groups which contain a regular subgroup. Heavily involved in our proofs are some new results on subgroups of simple groups which have orders divisible by various primes. For example, another corollary implies that for every simple group T apart from L 3 (3), U 3 (3), and L 2 ( p ) with p a Mersenne prime, there is a collection Π consisting of two or three odd prime divisors of | T |, such that if M is a subgroup of T of order divisible by every prime in Π, then | M | is divisible by all the prime divisors of | T |, and we obtain a classification of such subgroups M .
Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering
View Paper: https://www.semanticscholar.org/paper/c0b971cd5d00040aabcc1c27e9dddae70b22fbf2
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In Artificial Intelligence
In Discrete Mathematics and Combinatorics
Citations in the corpus
- REGULAR SUBGROUPS OF THE AFFINE GROUP AND RADICAL CIRCLE ALGEBRAS (2009, unknown): “In fact, if λ ∈ F and x , y , z ∈ V , then Liebeck et al. [4] have stated that the regular subgroups given by Heged¨us are examples of transitive subgroups that contain no nontrivial normal subgroup of the socle.”
- Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks (2016, background): “A) := A⋊ Aut(A). When Ais equal to an elementary abelian p-group (i.e. A∼= (Z/(p))n), Hol(A) is also known as the affine group of A. Regular subgroups of the affine group have been studied for example in [32, 29], mainly as subgroups of primitive permutation groups. The following result is an easy gener”