Rendiconti del Seminario Matematico della Università di Padova

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Volume 121, 2009, pp. 1–11
DOI: 10.4171/RSMUP/121-1

Knit Products of Some Groups and Their Applications

Firat Ateş[1] and A. Sinan Çevik[2]

(1) Department of Mathematics, Balikesir University, Campus, 10145, Balikesir, Turkey
(2) Department of Mathematics, Selçuk University, Campus, 42003, Konya, Turkey

Let G be a group with subgroups A and K (not necessarily normal) such that G = AK and AK = {1}. 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.

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Ateş Firat, Çevik A. Sinan: Knit Products of Some Groups and Their Applications. Rend. Sem. Mat. Univ. Padova 121 (2009), 1-11. doi: 10.4171/RSMUP/121-1