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Characterization of small polygroups by their fundamental groups

D. Heidari

Faculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran

E-mail Address: dheidari82@gmail.com

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and

B. Davvaz

Department of Mathematics, Yazd University, Yazd, Iran

E-mail Address: davvaz@yazd.ac.ir

Corresponding author.

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https://doi.org/10.1142/S1793830919500472Cited by:1 (Source: Crossref)

Abstract

In this paper, we consider polygroup $⟨ P, \circ, 1,^{- 1} ⟩$ $〈 P, \circ, 1,^{- 1} 〉$ and prove necessary and sufficient conditions such that $P$ $P$ is non-commutative. Then by using the Maple programming, we obtain all polygroups of order less than five up to isomorphism. In fact, we determine all 115 non-isomorphic polygroups of order less than five and characterize them by their fundamental groups, i.e., polygroups with same fundamental group, say $G$ $G$ , classifies in the class $[G]$ $[G]$ . Finally, we obtain that the fundamental groups of 94 polygroups are the trivial group. The numbers of polygroups in classes $[ℤ_{2}]$ and $[ℤ_{3}]$ are 16 and 3, respectively, and the classes $[ℤ_{2} \times ℤ_{2}]$ and $[ℤ_{4}]$ are singleton.

Keywords:

AMSC: 20N20

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