Faithful Action Research Articles

Speed’s Orbit Problem: Variations on Themes of Burnside

The paper offers a solution to T. P. Speed’s Orbit Problem, describing the orbit decomposition of direct powers of a transitive finite group action. The solution is given in terms of the Burnside algebra of the group and the right action of an incidence algebra of the subgroup lattice on a module of finitary functions. As a corollary, it is shown that almost all orbits in powers of a faithful action are regular. This gives one possible ‘GF(1)’ version of Burnside’s original theorem that each irreducible complex representation of a finite group appears in a tensor power of a fixed faithful representation.

INSURER EXPOSURE TO EXTRACONTRACTUAL LIABILITY UNDER STATE UNFAIR CLAIMS SETTLEMENT PRACTICES ACTS: AN EMPIRICAL TEST

Abstract: Litigation against insurers for bad faith actions has increased. The authors examine how the exposure of insurers has changed with particular emphasis on the impact of Unfair Claims Settlement Practices Acts.

A class of finitely generated groups with irrational translation numbers

In this paper we describe a class of examples of groups having word metrics with irrational translation numbers, namely groups of the form ℤ n ×φ, n ≥3 where $ {\mit\phi} \in {\rm \ GL} (n,{\Bbb Z}) $ is irreducible and has precisely two eigenvalues of unit modulus. These translation numbers can be computed explicitly. In addition, we prove that if the group G is of the form G=ℤ n ×γ where $ {\mit\gamma} \in {\rm \ GL} (n,{\Bbb Z}) $ , n ≥2 is irreducible and has an eigenvalue of unit modulus then G has a (indiscrete) cocompact faithful action by translations on $ {\Bbb R}^3 $ .

Symmetric (79, 27, 9)-designs Admitting a Faithful Action of a Frobenius Group of Order 39

In this paper we present the classification of symmetric designs with parameters (79, 27, 9) on which a non-abelian group of order 39 acts faithfully. In particular, we show that such a group acts semi-standardly with 7 orbits. Using the method of tactical decompositions, we are able to construct exactly 1320 non-isomorphic designs. The orders of the full automorphism groups of these designs all divide 8 · 3 · 13.

Book Review: The Gift of Anger: A Call to Faithful Action

Book Review: The Gift of Anger: A Call to Faithful Action

Torus Actions on Rings

We study torus actions on non-commutative rings, focusing on upper bounds on the dimensions of tori for which faithful actions are possible. We give sharp bounds for actions on algebras of generic matrices and their trace rings.

Non-commutative separability and group actions

We give conditions for the skew group ring S * G to be strongly separable and H-separable over the ring S. In particular we show that the H-separability is equivalent to S being central Galois extension. We also look into the H-separability of the ring S over the fixed subring R under a faithful action of a group G. We show that such a chain: S * G H-separable over S and S H-separable over R cannot occur, and that the centralizer of R in S is an Azumaya algebra in the presence of a central element of trace one.

Irreducible actions and faithful actions of hopf algebras

LetH be a Hopf algebra acting on an algebraA. We will examine the relationship betweenA, the ring of invariantsA H, and the smash productA # H. We begin by studying the situation whereA is an irreducibleA # H module and, as an application of our first main theorem, show that ifD is a division ring then [D : D H]≦dimH. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness ofA # H is strongly related to the faithfulness of the left and right actions ofA # H onA.

Actions of complex Lie groups on analytic ?-algebras

On a reduced analytic .ℂ-algebraR there are faithful analytic actions of complex Lie groups of arbitrarily high dimension if and only ifR has Krull dimension ≥2.

Torsion units in integral group rings of metacyclic groups

It is proved that if G is a split extension of a cyclic p-group by a cyclic p′-group with faithful action then any torsion unit of augmentation one of Z G is rationally conjugate to a group element. It is also proved that if G is a split extension of an abelian group A by an abelian group X with (| A|, | X|) = 1 then any torsion unit of Z G of augmentation one and order relatively prime to | A| is rationally conjugate to an element of X.

Monoid extension theory

The formal description familiar from group theory can be carried out in those monoid extensions whose congruence classes have representatives generating them under one-sided action of the kernel. This is first developed systematically, without further restrictive assumption. The original description of Rédei ‘9’ falls out when the kernel acts faithfully on these representatives. The “quasi-decompositions” and “triple sums” of Schmidt ‘11’ are seen to be the split extensions in the commutative case; and their generalization by Krishnan ‘5’ those in an (unnecessarily) symmetrized non-commutative one. (Split extensions under faithful action, alias “semidirect products”, also appear in ‘8, p. 186’ and more specially, with the kernel a group, in ‘10, p. 191’.) The “left coset” and “ H -coextensions” of Grillet and Leech ‘3, 7’ do not quite fit into the formulation but a slight modification covers them as well, while freeing them from restriction to a group “kernel”; thus liberated the construction applies, again back in the split commutative case, to yield Schmidt's “semigroup of semigroups”. p]An appendix extends recent work of Köhler to develop a Kaloujnine–Krasner theorem for these extensions.

Absolutely faithful group actions

The definition and main result. It has been shown ((1), § 3) that if G is any finite group and p any prime number not dividing |G|, then the number of conjugacy classes of maximal nilpotent subgroups in the regular wreath product of a cyclic group of order p by G is equal to the number of conjugacy classes of all nilpotent subgroups in G. This fact, together with various properties of the map by means of which it was established, proved helpful in dealing with questions of construction raised in (1). The present note isolates the key property of the wreath product on which the argument rests, and from this shows how the argument can be carried over to a more general context. The essential situation is that a group G acts on a group A in a way which will be called ‘absolutely faithful’.