Embedding theorems for ascending nilpotent groups

Abstract

In this paper two proofs will be given showing that an ascending nilpotent group can be embedded in the multiplicative group of a division ring, if and only if its subgroup of elements of finite order can be embedded in a division ring. The first proof will depend on the material in a paper by the author (see [3]), and the second proof will use the results in the first part of the paper On ordered division rings by B. H. Neumann (see [5]). The construction in the first proof is used to show that any automorphism of the group can be extended to the division ring. First we will determine which countably generated locally finite groups can be embedded in a division ring. As a corollary to this a criterion for embedding locally nilpotent periodic groups will be obtained. With this result we will be able to completely determine which ascending nilpotent groups can be embedded in a division ring. In this paper a group G will be said to have property E if it can be embedded in a division ring D, and to have property EE if every automorphism of the group G can be extended to be an automorphism of some division ring D. The theorem on countable locally finite groups depends mainly on a result of Amitsur (see [1]). The following notation is adapted from the notation of Amitsur's paper. 7r will denote the set of all primes and uri the set of all odd primes p such that 2 has odd order mod p. Let m and r be relatively prime integers. Put s = (r -1, m), t = m/s and n = minimal integer satisfying rn_ 1 mod m. Denote by Gm,,r a group generated by two elements A and B satisfying the relations Am= 1, Bn=At, and BAB-l=Ar. Denote by G:,, a group G which has an ascending tower of subgroups 1Hi| 0? i< oo } such that G = U=1 Hi and each Hi is isomorphic to Gmj,ri for some relatively prime integers mi and ri. Let si = (ri -1, m), ti = mi/si and ni = minimal integer satisfying rei-z 1 mod ni. T*, 0* and I* will denote the binary tetrahedral, octahedral, and icosahedral groups (see [1]). Amitsur proved the following: