Characterization of small polygroups by their fundamental groups
In this paper, we consider polygroup [Formula: see text] and prove necessary and sufficient conditions such that [Formula: see text] 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 [Formula: see text], classifies in the class [Formula: see text]. Finally, we obtain that the fundamental groups of 94 polygroups are the trivial group. The numbers of polygroups in classes [Formula: see text] and [Formula: see text] are 16 and 3, respectively, and the classes [Formula: see text] and [Formula: see text] are singleton.