Title:

Subnormality and soluble factorised groups

Throughout this summary the group G = AXB is always a product of three abelian subgroups A, X and B. In Chapter 1 we study finite 2groups G, where A and B are elementary and X has order 2. We also assume that X normalises both A and B, and thus AX and XB are nilpotent of class at most 2. We show that when the order of G divides 213 then G has derived length at most 3 ((1.4.2) and (1.6.1)). This supports the conjecture [see Introduction] on the derived length of a group which is expressible as the product of two nilpotent subgroups. In Chapter 2 we consider some special cases of G where A, X and B are finite pgroups and X is cyclic. We obtain a bound for the derived length of G which is independent of the prime p and the order of X. In Chapter 3 we find a bound for the derived length of a finite group G in terms of the highest power of a prime dividing the order of X when Ax = A, Bx = B and X is subnormal in both AX and XB. The most general result is Theorem (3.5.1). If G is a finite pgroup and X has order p we show that G has derived length at most 4 (Theorem (3.3.1)). Further in Chapter 3 if Ax « A, Bx = B, X < m AX and X < m XB then a bound for the subnormal defect of X in G is given. When X has order p this bound depends only upon m (see (3.3.4)), and when X has order pn and m is fixed then the subnormal defect of X in G can be bounded in terms of n (see the remark following Proposition (3.4.2)). Chapter 4 shows how some results from Chapters 2 and 3 can be generalised to infinite groups. Theorem (4.3.1) shows that when A and B are p groups of finite exponent, X has order pn, Ax = A, Bx = B, X < 2 AX and X < 2 XB then G is a locally finite group. Proposition (4.2.2) and Corollary (4.2.3) then enable some of the results about finite groups to be applied.
