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چکیده
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In this paper, we consider polygroup 〈P,∘,1,−1〉 and prove necessary and sufficient conditions such that 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, classifies in the class [G]. Finally, we obtain that the fundamental groups of 94 polygroups are the trivial group. The numbers of polygroups in classes [ℤ2] and [ℤ3] are 16 and 3, respectively, and the classes [ℤ2×ℤ2] and [ℤ4] are singleton.
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