A subgroup of the linear translation complement of a translation plane is geometrically irreducible if it has no invariant lines or subplanes. A similar definition can be given for "geometrically primitive". If a group is geometrically primitive and solvable then it is fixed point free or metacyclic or has a normal subgroup 2a+b a of order w where w divides the dimension of the vector space. Si...

Abstract Let G be a permutation group on set $\Omega $ of size t . We say that $\Lambda \subseteq \Omega is an independent if its pointwise stabilizer not equal to the any proper subset define height maximum set, and we denote this quantity $\textrm{H}(G)$ In paper, study for case when primitive. Our main result asserts either $\textrm{H}(G)< 9\log t$ or else in particular well-studied famil...

We show a method to effectively compute the Wedderburn decomposition and the primitive central idempotents of a semisimple finite group algebra of an abelian-by-supersolvable group G from certain pairs of subgroups of G. In this paper F = Fq denotes a finite field with q elements and G is a finite group of order n such that FG is semisimple, or equivalently (q, n) = 1. The group algebra FG is a...

Part I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O’Nan–Scott Theorem and Aschbacher’s theorem. Tables of the groups G are given for each O’Nan– Scott class. For the non-affine groups, additional information is given: the degree d of G, the shape of a stabiliser in G of the primitive action, the shape of the normali...

The relational complexity ρ(X,G) of a finite permutation group is the least k for which the group can be viewed as an automorphism group acting naturally on a homogeneous relational system whose relations are k-ary (an explicit permutation group theoretic version of this definition is also given). In the context of primitive permutation groups, the natural questions are (a) rough estimates, or ...