Some applications of tree-limits to groups. I

Regular and p-regular orbits of solvable linear groups

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

Regular orbits of cyclic subgroups in permutation representations of certain simple groups

Prime UV Systems Inc., Booth 1447

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic biprimitive finite groups saturated with groups of the sets {L2(pn)}, {Re(32n+1)}, and {Sz(22n+1)}. It is proved thai such groups are all isomorphic to {L2(P)}, {Re(Q)}, or {Sr(Q)} over locally finite fields.

The use of permutation representations in structural computations in large finite matrix groups

This paper is a survey of some of the ways in which the representation theory of the symmetric group has been used in voting theory and game theory. In particular, we use permutation representations that arise from the action of the symmetric group on tabloids to describe, for example, a surprising relationship between the Borda count and Kemeny rule in voting. We also explain a powerful representation-theoretic approach to working with linear symmetric solution concepts in cooperative game theory. Along the way, we discuss new research questions that arise within and because of the representation-theoretic framework we are using.

For a bifurcation germ F(x,λ):ℝn+1,0→ℝn,0 which is equivariant with respect to the action of a finite group G, there are permutation actions of G on various subsets of branches of F−1(0). These sets include the set of all branches as well as the set of branches where λ>0 or 0 or <0, We shall give formulas for the modular characters of these permutation representations (which are the regular characters restricted to the odd order elements of G). These formulas are in terms of the representations of G on certain finite dimensional algebras associated to F. We deduce sufficient conditions for the existence of submaximal orbits by comparing the permutation representations for maximal orbits with certain representations of G.

A permutation representation of a finite group is multiplicity-free if all the irreducible constituents in the permutation character are distinct. There are three main reasons why these representations are interesting: it has been checked that all finite simple groups have such permutation representations, these are often of geometric interest, and the actions on vertices of distance-transitive graphs are multiplicity-free. In this paper we classify the primitive multiplicity-free representations of the sporadic simple groups and their automorphism groups. We determine all the distance-transitive graphs arising from these representations. Moreover, we obtain intersection matrices for most of these actions, which are of further interest and should be useful in future investigations of the sporadic simple groups.

Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not necessarily minimal) permutation representations P . It is unusual, but significant to recognize that a P is a Grothendieck’s “dessin d’enfant” D and that a wealth of standard graphs and finite geometries G —such as near polygons and their generalizations—are stabilized by a D . In our paper, tripods P − D − G of rank larger than two, corresponding to simple groups, are organized into classes, e.g., symplectic, unitary, sporadic, etc. (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All of the defined geometries G ′ s have a contextuality parameter close to its maximal value of one.

Regular orbits of sporadic simple groups

<p style='text-indent:20px;'>In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula> action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, the centralizer of any minimal aperiodic <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-action on the Cantor set is a subgroup of the centralizer of a minimal <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula>-action.

Inversion tables and minimal left coset representatives for Weyl groups of classical type

The permutation representations in the title are all determined, and no surprises are found to occur.

Spherical 5-designs obtained from finite unitary groups