Group Rings with Solvable «-engel Unit Groups'

Let KG be the group ring of a group G over a field of characteristic p > 0, p ^ 2, 3. Suppose G contains no element of order p (if p > 0). Group algebras KG with unit group U(KG) solvable and n-Engel are characterized. Let ATG be the group ring of a group G over a field K of characteristic p > 0 and let U(KG) denote its group of units. Several authors including Bateman [1], Bateman and Coleman [2], Motose and Tominaga [10] and Khripta [5] have studied the question as to when U(KG) is solvable or nilpotent. Khripta in a beautiful paper [5] has proved that if p > 0 and G has a /7-element then U(KG) is nilpotent if and only if G is nilpotent and the derived group C is a finite /?-group, settling the nonsemiprime case. This, incidently, is equivalent to saying that KG is Lie nilpotent (see [11] and [14]). Khripta also has some results in her thesis on the nilpotency of U(KG) in the semiprime case. We investigate when U(KG) is a solvable n-Engel group; more precisely we prove Theorem. Suppose KG is a group ring over a field K of characteristic p > 0, p=£2,3. Suppose G has no element of order p (if p > 0). Then the following are equivalent. (i) U(KG) is solvable and n-Engel. (ii) G is solvable and m-Engel and one of (a), (b) holds. (a) T(G), the set of torsion elements of G, is central in G. (b) \K\ = 2P — \ = p, a Mersenne prime; T(G) is abelian of exponent (p2 1) and for x G G, t G T(G), xt # tx => x" vx = tp. (iii) U(KG) is nilpotent. We are indebted to the referee for several useful comments. 1. Notations and definitions. For group elements x, y we write the commutator (x, y) = x>>x ~ ly ~' and (x, y,y, . . . , v-) = (x, y,^^y)y{x, y, . . . ,y) y~\ A group H is «-Engel if it satisfies (x, y, . . . ,y\ = I for all x,y G H n Received by the editors December 4, 1974. AMS (MOS) subject classifications (1970). Primary 16A26; Secondary 20F45.