We construct the first infinite families of locally 2-arc transitive graphs with the property that the automorphism group has two orbits on vertices and is quasiprimitive on exactly one orbit, of ...twisted wreath type. This work contributes to Giudici, Li and Praeger’s program for the classification of locally 2-arc transitive graphs by showing that the star normal quotient twisted wreath category also contains infinitely many graphs.
Let L be the set of all Fitting classes F with the following two properties: (i)$\mathscr{F} \supseteq \mathscr{R}$, the class of all finite nilpotent groups, and (ii) every F-avoided, complemented ...chief factor of any finite soluble group G is partially F-complemented in G. It is shown that L is a complete sublattice of the complete lattice N of all nontrivial normal Fitting classes, and, moreover, it is lattice isomorphic to the subgroup lattice of the Frattini factor group of a certain abelian torsion group due to H. Lausch.
We study the base sizes of finite quasiprimitive permutation groups of twisted wreath type, which are precisely the finite permutation groups with a unique minimal normal subgroup that is also ...non-abelian, non-simple and regular. Every permutation group of twisted wreath type is permutation isomorphic to a twisted wreath product G=Tk:P acting on its base group Ω=Tk, where T is some non-abelian simple group and P is some group acting transitively on k={1,…,k} with k⩾2. We prove that if G is primitive on Ω and P is quasiprimitive on k, then G has base size 2. We also prove that the proportion of pairs of points that are bases for G tends to 1 as |G|→∞ when G is primitive on Ω and P is primitive on k. Lastly, we determine the base size of any quasiprimitive group of twisted wreath type up to four possible values (and three in the primitive case). In particular, we demonstrate that there are many families of primitive groups of twisted wreath type with arbitrarily large base sizes.
A permutation group G ≤ Sym(X) on a finite set X is sharp if |G|=∏
l∈L(G)
(|X| − l), where L(G) = {|fix(g)| | 1 ≠ g ∈ G}. We show that no finite primitive permutation groups of twisted wreath type ...are sharp.
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