2-closures of primitive permutation groups of holomorph type

Closures of Finite Primitive Permutation Groups

Let G be a primitive permutation group on a finite set ft, and, for k ^ 2, let G be the Ar-closure of G, that is, the largest subgroup of Sym (ft) preserving all the G-invariant ^-relations on ft. Suppose that G<H^ G and G and H have different socles. It is shown that k ^ 5 and the groups G and H are classified explicitly.

Primitive Permutation Groups of Simple Diagonal Type

Let G be a finite primitive group such that there is only one minimal normal subgroup M in G, this M is nonabelian and nonsimple, and a maximal normal subgroup of M is regular. Further, let H be a point stabilizer in G. Then H (1 M is a (nonabelian simple) common complement in M to all the maximal normal subgroups of M, and there is a natural identification ofMwith a direct power T m of a nonab...

QUASI-PERMUTATION REPRESENTATIONS OF METACYCLIC 2-GROUPS

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus, every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fa...

Factorizations of Primitive Permutation Groups

Distinguishing Primitive Permutation Groups

Let G be a permutation group acting on a set V . A partition π of V is distinguishing if the only element of G that fixes each cell of π is the identity. The distinguishing number of G is the minimum number of cells in a distinguishing partition. We prove that if G is a primitive permutation group and |V | ≥ 336, its distinguishing number is two.