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Embedding Groups into Distributive Subsets of the Monoid of Binary Operations
by
Gregory Mezera
George Washington University

Let X be a set and Bin(X) the set of all binary operations on X. We can make Bin(X) into a semigroup with the operation x*₁*₂y=(x*₁y)*₂y for all *1, *2 in Bin(X) and all x, y in X. We say that S ⊂ Bin(X) is a distributive set of operations if all pairs of elements *_α, *_β ∈ S are right distributive, that is, (a*_αb)*_βc = (a*_βc)*_α(b*_βc) (we allow *_α=*_β). J.Przytycki raised the question of which groups can be realized as distributive sets. The initial guess that we may embed any group G into Bin(X) for some X was brought into question after Michal Jablonowski made an observation that if * ∈ S is idempotent (a*a=a), then * commutes with every element of S. In addition, Agata Jastrzebska computed all groups embedded in Bin(X) with |X| < =5, and found no nonabelian groups. However, the first noncommutative subgroup of Bin(X) (the group S₃) was found computationally in October of 2011 by Yosef Berman. Here we show that any group can be embedded in Bin(X) for X=G (as a set). We do this by giving an explicit embedding we call the regular embedding, due to its relation with the regular representation of G. This embedding sends g to *_g where

a*gb=ab-1gb.

Here, we check that the group {*_g}_{g ∈ G} is a distributive set: We have:

(a*g1b)*g2c = (ab-1g1b)*g2c = ab-1g1bc-1g2c and

(a*g2c)*g1(b*g2c)=(ac-1g2c)*g1(bc-1g2c) = ab-1g1bc-1g2c as needed.

We also discuss criteria for minimal embeddings of finite groups. That is, for a given group G, we show that the minimal |X| such that G embeds into a distributive subset of Bin(X) is related to the degree of the minimal faithful representation of G over F₂, the field of two elements.

Paper reference: http://front.math.ucdavis.edu/1109.4850

Date received: March 9, 2012

Copyright © 2012 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbek-13.