Department of Mathematics
Server

In 1942, Levi proved that a group satisfies the commutator law \([[x, y], z]= [x, [y, z]]\) if and only if the group is of nilpotent of class at most 2 (see A. G. Kurosh for a modern proof). By a well-known result of Neumann and Taylor (also, independently by Mal'cev), a cancellation semigroup satisfies the positive semigroup law \( xyzyx = yxzxy \) if and only it is a subsemigroup of a group of nilpotent class at most 2. This situation clearly calls for a conjugacy analogs of Levi's theorem for groups and semigroups. In other words, is there an identity in the language of one binary operation \( x@y \) (= \( y^{-1}xy \) in groups) characterizing groups of nilpotent class at most 2. The associativity of the binary operation of conjugacy will be too powerful in this context. In fact, it will force the semigroup to be commutative. Here we prove an analog of Levi's theorem for conjugates by characterizing semigroups embeddable in groups of nilpotent class at most 2 by means of a single conjugacy law. This result is new even for groups.

Theorem.
In a cancellative semigroup \( S \) admitting conjugates, the following two statements are equivalent:

1. \(S \) satisfies the conjugacy law \( x@(y@z) = x@y \)
   (In standard notation \( x^{(y^z)} = x^y \)).
2. \( S\) is embeddable in a group of nilpotent class at most 2.

1. Levi, F. W., Groups in which the commutator operation satisfies certain algebraic conditions J. Indian Math. Soc. (N.S.) 6, 1942, 87 - 97.
2. Kurosh, A. G., The Theory of Groups, Volume I, Chelsea, New York, 1956.
3. Moghaddam, G. I., On Semigroups admitting Commutators or Conjugates, CMS Conference, Winnipeg 2014.
4. Neumann, B.H.; Taylor, T; Subsemigroups of nilpotent groups, Proc. Roy. Soc, Ser. A 274, 1963, 1 - 4.
5. Padmanabhan, R. , Algebra and Geometry with Computers, CMS Conference, Winnipeg 2014.