Locally finite and solvable subgroups of sfields

In this paper we will determine all locally finite subgroups of a sfield and give a criterion for determining when the group algebra of an ascending solvable torsion free group has a quotient sfield. Notation and Definitions.A group G has property E if it can be embedded in a sfield D and property EE if every automorphism of the group G can be extended to be an automorphsim of D. For a more complete discussion see [5].The result on locally finite groups depends mainly on a result of Amitsur (see [l]).A complete discussion of the following notation can be found in his paper.■k will denote the set of all primes and 7Ti 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 « = minimal integer satisfying rn=l mod m.Denote by Gm,r a group generated by two elements A and B satisfying Am = \, Bn = Al and BAB~l = Ar.Denote by G", a group G which has a countable ascending tower of subgroups {Hí:0^í< oo j such that G = U°Li Hi and each Hi is isomorphic to Gmi,Ti.T*, 0*, I* will denote the binary tetrahedral, octahedral and icosahedral groups.Let p be a fixed prime dividing m. a = ap is the highest power of p dividing m. i]p is the minimal integer satisfying r^=\ mod (mp~a).pp is the minimal integer satisfying r"» = £" mod (mp~a) for some integer p'.Sj, is the minimal integer such that ps"=l mod (mp~a).o'P=pp8p/r¡p.Condition C. Integers m and r satisfy Condition C if either (I) (re, t) = (s, 0 = 1, or (II) re = 2re', m = 2am', s = 2s' where a^2, m', s' and re' are odd integers; (re, t) = (s, t)=2 and r= -1 mod 2".