‎we investigate two constructions‎ - ‎the replacement and the zig-zag‎ ‎product of graphs‎ - ‎describing several fascinating connections‎ ‎with combinatorics‎, ‎via the notion of expander graph‎, ‎group‎ ‎theory‎, ‎via the notion of semidirect product and cayley graph‎, ‎and‎ ‎with markov chains‎, ‎via the lamplighter random walk‎. ‎many examples‎ ‎are provided‎.

In this paper, we introduce a new product operation of association schemes in order to generalize the notion of semidirect products and wreath product. We then show that our construction covers some association schemes which are neither wreath products nor semidirect products of two given association schemes.

In his 1996 work developing the theory of association schemes as a ‘generalized’ group theory, Zieschang introduced the concept of the semidirect product as a possible product operation of certain association schemes. In this paper we extend the semidirect product operation into the entire set of association schemes. We then derive a way to decompose certain association schemes into smaller ass...

A notion of graph-wreath product of groups is introduced. We obtain sufficient conditions for these products to satisfy the topologically inspired finiteness condition type Fn. Under various additional assumptions we show that these conditions are necessary. Our results generalize results of Bartholdi, Cornulier and Kochloukova about wreath products. Graphwreath products of groups include class...

The wreath product of groups A B is one of basic constructions in group theory. We construct its analogue, a wreath product of Lie algebras. Consider Lie algebras H and G over a field K. Let U(G) be the universal enveloping algebra. Then H̄ = HomK(U(G), H) has the natural structure of a Lie algebra, where the multiplication is defined via the comultiplication in U(G). Also, G acts by derivations...

In this paper we describe $${\mathcal {R}}$$ -unipotent semigroups being regular extensions of a left band by an $$\mathcal {R}$$ semigroup T as certain subsemigroups wreath product T. We obtain Szendrei’s result that each E-unitary is embeddable into semidirect group. Further, specialising the first author’s notion $$\lambda $$ -semidirect locally semigroup, provide answer to open question rai...

This paper is concerned with finite primitive permutation groups G which are subgroups of wreath products W in product action and are such that the socles of G and W are the same. The aim is to explore how the study of such groups may be reduced to the study of smaller groups. The O'Nan-Scott Theorem (see Liebeck, Praeger, Saxl [12] for the most recent and detailed treatment) sorts finite primi...

Let X = (X, H) and Y = (Y, K) be two association schemes. In [1], an (external) semidirect product Z = (X × Y, H nπ K) of Y and X relative to π is defined. In this paper, given a normal closed subset N of K which is fixed by π, we find a way to construct a fusion scheme of Z with respect to N . We then show that every association scheme obtained by any of three products, direct, wreath and semi...

We introduce semidirect and wreath products of finite ordered semigroups and extend some standard decomposition results to this case.

we regard the shearlet group as a semidirect product group andshow that its standard representation is,typically, a quasiregu-lar representation. as a result we can characterize irreducibleas well as square-integrable subrepresentations of the shearletgroup.