Groups with identities
This paper is concerned with groups which satisfy a finite ``monoidal disjunction'' \(\bigvee^{n}_{i=1}w_ i(x,y)=v_ i(x,y)\) where, for each i, \(v_ i\) and \(w_ i\) are distinct elements of the free monoid on x and y. It was shown by \textit{M. Boffa} [C. R. Acad. Sci., Paris, Sér. I 303, 587-589 (1986; Zbl 0619.03026)] that these groups may also be described as those admitting the elimination of inverses (e.i.). The author defines what it means for a group to admit e.i. with complexity \(\leq \ell\) (where \(\ell\) is a positive integer). It is immediate from the definition that every group which admits e.i. must admit e.i. with complexity \(\leq \ell\) for some \(\ell\). One main result of the paper is that if G is a soluble group which admits e.i. then G is uniformly locally quasi-nilpotent: in other words, for every k there exist c and i such that every k-generator subgroup of G has a nilpotent subgroup of class c and of finite index i. Moreover, if G admits e.i. with complexity \(\leq \ell\) then c can be computed explicitly in terms of k, \(\ell\) and the solubility length of G. Another main result (using the classification of finite simple groups) is that if G is a locally finite group which admits e.i. then G has a locally nilpotent normal subgroup N such that G/N has finite exponent.
scientific article; zbMATH DE number 4142296
scientific article