After precalculus, mathematics tudents often leave behind the familiar family of transformations, x Fax + b. We will show that this family, when examined in the right light, leads us to some interesting and important ideas in group theory. By building on this accessible example, it is possible to introduce the semidirect product (a topic usually first seen at the graduate level) in an undergrad...

We present a group theoretic construction of the Virasoro algebra in the framework of wreath products. This can be regarded as a counterpart of a geometric construction of Lehn in the theory of Hilbert schemes of points on a surface. Introduction It is by now well known that a direct sum ⊕ n≥0R(Sn) of the Grothendieck rings of symmetric groups Sn can be identified with the Fock space of the Hei...

We give an algebraic characterization of the tree languages that are defined by logical formulas using certain Lindström quantifiers. An important instance of our result concerns first-order definable tree languages. Our characterization relies on the usage of preclones, an algebraic structure introduced by the authors in a previous paper, and of the block product operation on preclones. Our re...

We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals. We give a semidirect product decomposition for an abelian groupoid. This is done through a matched pair and leads to a C*-diagonal (for a special case). We use this decomposition to study ...

Let G be a group acting faithfully on a set X . The distinguishing number of the action of G on X , denoted DG(X ), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a colorpreserving permutation of X . In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. ...

In this paper, we study the subconstituent algebras, also called as Terwilliger algebras, of association schemes that are obtained as the wreath product of one-class association schemes Kn = H(1, n) for n ≥ 2. We find that the d-class association scheme Kn1 o Kn2 o · · · o Knd formed by taking the wreath product of Kni has the triple-regularity property. We determine the dimension of the Terwil...