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A well-known theorem of Hanna Neumann says that the only binary group terms which are associative in all groups are \(x\), \(y\), \(xy\), and \(yx\). The same statement is clearly valid for abelian groups as well. In 1942, F. W. Levi proved that the commutator operation in a group - viewed as a binary function - is associative if and only if the group is nilpotent of class at most 2. Here we prove a very similar theorem for the derived binary term \(y^{-1}xy^{2}\): the binary law \(y^{-1}xy^{2}\) in a group is associative if and only if the group is of nilpotent class at most 2. We also formulate and prove a new semigroup generalization of this theorem.

This work is done in collaboration with Yang Zhang.

References:

1. A.H. Copeland and A.O. Shar, Constructing loops from groups, Amer. Math. Monthly 88 (1981), no.4,283-285
2. F.W. Levi, Groups in which the commutator operation satisfies certain algebraic conditions., J. Indian Math. Soc. (N.S.) 6 (1942), 87-97
3. H. Neumann, On a question of Kertész, Publ. Math. Debrecen 8 (1961), 75-78
4. W. McCune. R. Padmanabhan and R. Veroff;, Levi's commutator theorems for cancellative semigroups, Semigroup Forum 71 (2005), 152-157.