Knit Products of Some Groups and Their Applications

Abstract

Let G be a group with subgroups A and K (not necessarily normal) such that G AK and A K f1g. Then G is isomorphic to the knit product, that is, the``two-sided semidirect product'' of K by A. We note that knit products coincide with Zappa-Szep products (see [18]). In this paper, as an application of [2, Lemma 3.16], we first define a presentation for the knit product G where A and K are finite cyclic subgroups. Then we give an example of this presentation by considering the (extended) Hecke groups. After that, by defining the Schur multiplier of G, we present sufficient conditions for the presentation of G to be efficient. In the final part of this paper, by examining the knit product of a free group of rank n by an infinite cyclic group, we give necessary and sufficient conditions for this special knit product to be subgroup separable.

Some algebraic structures on the generalization general products of monoids and semigroups

“…Define an action of B ⊕A on A ⊕B by f g = ( f g) + and an action of A ⊕B on B ⊕A by f g = f g. Then A ⊕B δ ψ B ⊕A is the generalized general product of B ⊕A and A ⊕B . Proof To proof this lemma, we need to check these two actions whether they actually satisfy the properties defined in (1).…”

“…The notion of Zappa-Szép products generalizes those of direct and semidirect products; the key property is that every element of the Zappa-Szép product can be written uniquely as a product of two elements, one from each factor, in any given order. In the literature, there are some key stone studies on the general product which is also referred as bilateral semidirect products (see [11]), Zappa products (see [7,12,16,18]) or knit products (see [1,14]). As a next step of general product, in [4], the same authors of this paper have recently introduced the generalization of the general product under the name of a higher version of Zappa products for monoids as in the following:…”

Commutator subgroups of generalized Hecke and extended generalized Hecke groups,II

“…Notice that the generalized Hecke group H (2,3) is the modular group Γ = P SL (2, Z). The modular group is the discrete subgroup of P SL(2,R) generated by two linear fractional transformations…”

A Higher Version of Zappa Products for Monoids

“…Nevertheless, direct, semidirect and (standard) wreath products are the most famous structures among these extension constructions (see, for instance, [10,14,18,20,25]). As a next step of these products, some other people also studied Zappa (or Zappa-Szép) products ( [13,16,27,28]) which is also referred as bilateral semidirect products ( [22]), general products ( [23]) or knit products ( [1,26]). Unlikely semi-direct products, none of the factor is normal in the Zappa product of any two groups.…”