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In this paper we define a new monoid construction under crossed products for given monoids. We also present a generating set and a relator set for this product. Finally, we give the necessary and sufficient conditions for the regularity of it. MSC: 05C10; 05C12; 05C25; 20E22; 20M05

In this paper, we generalize some results related to the congruence subgroups of modular group ; given in [7] and [6] by Kiming, Schütt, and Verrill, to the extended modular group :

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We consider the extended Hecke group [Formula: see text] generated by T(z)=-1/z, [Formula: see text] and [Formula: see text] with [Formula: see text]. In this paper, we study the abstract group structure of the extended Hecke group and its power subgroups [Formula: see text]. Also, we give relations between power subgroups and commutator subgroups of the extended Hecke group [Formula: see text].

A compact bordered Klein surface of algebraic genus p ≥ 2 has at most 12(p-1) automorphisms. Automorphism groups which attain this bound are called M*-groups. In this paper, firstly, we define generalized M*-groups. Then, we show that there is a relationship between the extended Hecke groups and generalized M*-groups. Finally, we prove that a generalized M*-groups G is supersoluble if and only if |G| = 4 · qr for q ≥ 3 prime number and for some positive integer r.