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A group \( G \) is \( n \)-abelian if the power-map \( f(x) = x^n \) is an endomorphism of the group. Thanks to a powerful structure theorem due to Reinhold Baer, Alperin, Kaluzhnin and others it is known that every torsion-free \( n \)-abelian group is abelian. Here, we generalize this theorem for cancellation semigroups i.e. a torsion-free \( n \) -abelian cancellation semigroup is commutative. Our proof is of first-order logic and we avoid using terms like \( x^n \) involving integer variables.. Instead we use power-like functions. This way, first-order theorem provers can prove this generalized version of the original group theory statement. In such an automated proof, the length of the proof as well as the length of the longest clause in the proof remain the same however huge the integer \( n \) may be.

This is a joint-work done in collaboration with Hossein Moghaddam and Yang Zhang.

1. Alperin, J. L., A classification of \( n \)-abelian groups, Canad. J . Math 21 (1969) 1238-1244.
2. Baer, R., Factorization of \( n \)-soluble and \( n \)-nilpotent groups, Proc. Amer. Math. Soc., 4, 15-26, (1953).
3. Kaluzhnin, L. A., The structure of \( n \)-abelian groupsÓ, Mat. Zametki, 2:5 (1967), 455-464.
4. Kurosh, A. G., The theory of groups, volumes 1-2, Chelsea, 1955-56.
5. Moghaddam, G.I and Padmanabhan, R., Commutativity theorems for cancellative semigroups (to appear in Semigroup Forum, 2016)
6. Padmanabhan, R, and Zhang, Yang., Automated Deduction in Ring Theory, ICMS Conference, Berlin, 2016.