1. Finite presentability of generalized Bruck–Reilly ∗-extension of groups
- Author
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Eylem Güzel Karpuz, Seda Oğuz, and [Oguz, Seda] Cumhuriyet Univ, Fac Educ, Dept Sch Sci & Math Educ 2, TR-58140 Sivas, Turkey -- [Karpuz, Eylem G.] Karamanoglu Mehmetbey Univ, Kamil Ozdag Sci Fac, Dept Math, Yunus Emre Campus, TR-70100 Karaman, Turkey
- Subjects
General Mathematics ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,Morphism ,Computer Science::General Literature ,Finitely-generated abelian group ,0101 mathematics ,Algebra over a field ,Identity element ,Finite set ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Discrete mathematics ,Group (mathematics) ,Computer Science::Information Retrieval ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Extension (predicate logic) ,Generalized Bruck-Reilly extension ,finite presentability ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,010201 computation theory & mathematics ,finite generation ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING - Abstract
WOS: 000389311800021, In [Finite presentability of Bruck-Reilly extensions of groups, J. Algebra 242 (2001) 2030], Araujo and RuAtic studied finite generation and finite presentability of Bruck Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck Reilly s -extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck Reilly s-extension of a group, G R. (G; beta, gamma; u), to be finitely generated and finitely presented. Let G be a group, (3,-y: be morphisms and u is an element of H-1 (H-1(*) and H-1 are the ?-G- and ?-t-classes, respectively, contains the identity element kr of T). We prove that GE R* (G; 3, 7; u) is finitely generated if and only if there exists a finite subset Xo C G such that G is generated by ((boolean OR(i >= 0) X-0,beta(i))U(Uj >= 0X0 gamma(j)). We also prove that C superset of R.* (C; beta, gamma; u) is finitely presented if and only if G is presented by (X; R), where X is a finite set and R = (U i >= 0 R-0 beta(i)) boolean OR (U-j >= 0 R0 gamma(j)) = {w(1)beta(i) = upsilon(1)beta(i) : i >= 0, w(1) = v(1) is an element of H-1* for some finite set of relations R0 subset of X* x X*.
- Published
- 2016