Groups with identities

Francçoise Point
1989 Annals of Pure and Applied Logic  
We characterize the soluble groups which admit a disjunction of monoidai identities. This notion has been introduced by An. Boffa and is equivalent to admitting the elimination of inverses. We use the result of Rosenblatt that a finitely generated soluble group which does not contain the free monoid on two generators is quasi-nilpotent. We also obtain partial results for the class of elementary amenable groups and its subclass of locally finite groups. Milnor and Wolf have shown that a finitely
more » ... generated soluble group with polynomial growth has a normal nilpotent subgroup of finite index. Rosenblatt has extended this result, noting that one can weaken the hypothesis of being of polynomial growth to that of not containing M2, the free monoid on two generators. Later Thou generalized senblatt's result to the finitely generated elementary amenable groups (see Section 3). We try to characterize the groups G which have the property that none of their elementary extensions contains A&. It is easy to see that this property is equivalent to G satisfying a finite disjunction of monoidal identities; it is also equivalent to G admitting elimination of inverses (e.i., in short) (see Section 1). We introduce the notion of having e.i. with complexity 4. If a group has e.i., it has e.i. with complexity dl, for some 1. show that a soluble group G which admits e.i. with complexity sI, is uniformly locally quasi-nilpotent, i.e. for every k, there exist c and i such that any subgroup generated by k elements has a nilpotent subgroup of class SC and of index <i; moreover, can be computed explicitely in terms of k, I and the solubility class of 6. have been unable to compute explicitly the index i (see Section 2). en we try to generalize this result for the elementary amenable groups. n two results in that direction. roperty of being unifo ly locally S. @I 1989, Elsevier Science Publishers 172 F. Point far as the direct unions are concerned, in collaboration with 6. Cherlin, we that a locally finite group which admits e.i. with complexity <I is (locally nent). The exponent can be calculated as a function to characterize nt groups with e.i. ent concerns the fact that of groups having e.i. onoidal identity. We do not know if this is an accident very special monoidal idertities obtained from the up G admits elimination of inverses (e-i.) iff every open formula in the -l, I) of groups is equivalent to an open formula in the language of An analogous n has been introduced for the division rings and those which t e.i. are exactly the division rings finite dimensional over their center [4]. algebraic result used in [4], is the theorem of Kaplansky that the division which satisfy a polynomial identk j-are exactly the finite dimensional ones over their center. introduce the notion of monoidal identity. A monoidal identity is an ty of the form cu(x, y) = /3(x, y), w h ere cu(x, y) and /3(x, y) are two distinct rds construeted on the alphabet {x, y}. ffa ]2]). A group G admits e.i. a disjunction of nwnoidal identities, iff any elementary extensbn of G does rwt contain , the fbee monoid on two genemtom. 0 exponent. Q 121). The property of having e.i. is preserved under otphic images and extemions by a group of finite be two groups wirkb do not denote by 6" the direct product of 6, o times with g noted the following: e.i. SQ
doi:10.1016/0168-0072(89)90060-2 fatcat:2j6yfeugffbt5k7ezha4gpas4q