| Abstract:
Recall that a group \( G \) is \( n \)-abelian if the power-map
\( f(x) = x^n \) is an endomorphism of the group.
The following theorems are well-known in group theory:
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\( n \)-abelian torsion-free groups are abelian (Baer, Alperin and others; [1], [2], [3])
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\( n \)-abelian implies that the group is \( n(n-1) \)-central
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\( (n, n+1, n+2) \)-abelian implies that the group is abelian
Here we formulate \( n \)-free versions of the above
and give first-order proofs of the resulting theorems for cancellative semigroups.
This is a joint-work done in collaboration with Hossein Moghaddam and Yang
Zhang ([4], [5]).
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Alperin, J. L., A classification of \( n \)-abelian groups,
Canad. J . Math 21 (1969) 1238-1244.
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Baer, R., Factorization of \( n \)-soluble and \( n \)-nilpotent groups,
Proc. Amer.
Math. Soc., 4, 15-26, (1953).
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Kaluzhnin, L. A., The structure of \( n \)-abelian groupsÓ,
Mat. Zametki,
2:5 (1967), 455-464.
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Moghaddam, G.I and Padmanabhan, R., Commutativity theorems for
cancellative semigroups (to appear in
Semigroup Forum, 2016)
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Padmanabhan, R, and Zhang, Yang., Automated Deduction in Ring Theory,
ICMS Conference, Berlin, 2016.
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