QE nil-2 groups of exponent 4

Abstract

The notion of a QE (quantifier-eliminable) group comes from logic: a group G is QE if every formula of the first-order language of group theory is equivalent in G to a formula without quantifiers. In the context of uniformly locally finite (ULF) groups, this logical property has an interesting algebraic equivalent: a ULF group G is QE if and only if every isomorphism between finite subgroups of G is induced by an automorphism of G. In this paper we show that there exist 2No countable ULF QE groups. It is well-known, and easy to verify, that every ULF QE group is Nocategorical. The existence of 2Ko countable &categorical groups is a recent result of Berline and Cherlin [ 11. Actually, Berline and Cherlin construct, for any prime p, 2No countable QE nilrings of characteristic p (necessarily Nocategorical). Using the Mal’cev correspondence [5] they then obtain, for every odd p, 2”” countable &-categorical nonabelian nil-2 groups of exponent p. However, it is easy to see that these groups are not QE, and in fact it can be shown [2] that every QE nonabelian nilpotent p-group must have exponent 4. The result we prove here is: