On the Symmetric Genus of K-metacyclic Group and Its Presentations
G. N. Shuaibu
Department of Mathematics and Statistics, University of Maiduguri, Nigeria.
B. A. Modu
Department of Mathematics and Statistics, University of Maiduguri, Nigeria.
D. Samaila *
Department of Mathematics, Adamawa State University, Mubi, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
The symmetric genus of a finite group G is defined as the smallest genus s of a compact connected oriented surface on which G acts faithfully via diffeomorphisms, which may be orientation-preserving or orientation-reserving. In this paper we determine a useful lower bound for the symmetric genus of any finite group with a cyclic quotient group. We examined the lower bound for the family of K-metacyclic groups and then determined the symmetric genus of each non-Abelian subgroup of a K-metacyclic group. We also provide some examples of groups for which the lower bound is attained and then used the standard presentation of a finite group as a quotient of a Non-Euclidean Crystallographic (NEC) group by a Fuchsian surface group.
Retraction Notice: This paper has been retracted from the journal after receipt of written complains. This journal is determined to promote integrity in research publication. This retraction is in spirit of the same. After formal procedures editor(s) and publisher have retracted this paper on 10th May-2019. Related policy is available here: http://goo.gl/lI77Nn
Keywords: Finite group, metacyclic group, symmetric genus, Riemann surface, representations, non-Euclidean crystallographic group, automorphism group.