Math. Z. 167, 37-47 (1979) Mathematische Zeitschrift 9 by Springer-Verlag 1979 The Join of Two Fitting Classes Elspeth Cusack School of Mathematics and Physics, University of East Anglia, Norwich, England Introduction All groups considered here are finite and soluble. A Fitting class ~ [-8] is a class of groups which is closed under taking normal subgroups and forming normal products of groups in ~ It is easy to show that the intersection of a family of Fitting classes is again a Fitting class. We may therefore define the join, or least upper bound, X v Y / o f two Fitting classes X and Y/to be the intersection of all those Fitting classes which contain their union. Let X and Y/be Fitting classes. We conjecture that a group lies in X v Y/if and only if it can be subnormally embedded in some group G, where G possesses normal subgroups H and K such that H~X, KEY/and G = H K . It is clear that each such group lies in X v Y/. It is far from clear, however, that the set of all such groups is a Fitting class. In Sect. 2 of this paper a number of theorems are proved which describe the join of Fitting classes X and Y/under various hypotheses on X and Y/. In each case the conjecture is found to have a positive answer. For example, we show in Corollary 2.6 that if Y/___X*, then a group G lies in X v Y/if and only if there exists a group K e y / s u c h that (G x K)~ is subdirect in G x K. Further, X v Y/is the set of such groups G only if Y/~X*. (If N is a normal subgroup of G x K, then N is subdirect in G x K if G is the image of N under the natural projection of N onto G, and K is the image of N under the natural projection of N onto K). Let X, Y/and ~" be Fitting classes such that X ~ ~ and our conjecture is true for X and Y/. As a consequence of the Dedekind Identity, we prove in Theorem 2.9 that (X v Y/) c~W = X v (Y/~ ~ ) . If N(Y/) denotes the smallest normal Fitting class containing a given Fitting class Y/, and 5~, denotes the smallest normal Fitting class [3], then we deduce from the last-mentioned result that X* c~5~, = W, if and only if X*c~ N(Y/)= Y/for each Fitting class Y/such that Y/*--X*. (Corollary 3.7 of [4]). In Sect. 3 we use an idea of P. Hauck to gain information about certain Fitting class joins. Finally, Sect. 4 contains some results on Fitting class joins which are in a similar vein to the theorems on Fitting class products in [1]. We consider normal Fitting 0025- 5874/79/0167/0037/$02.20 38 E. Cusack classes X which have the property that each Fitting class ~/such that X v Y/= 5 P is also normal. That this property is non-trivial is shown by our result that if X is a normal Fitting class, and there exists a positive integer n such that for each group G, IG/GxI< n, then there exists a non-normal Fitting class ~ with W v ~ = 5~. The preliminary results and definitions which will be needed are given in Sect. 1, and we use the following standard notation: FitT: the smallest Fitting class containing a set of groups T. ~: the class of all finite soluble groups. ~: the class of all finite nilpotent groups. Y~: the class of all finite soluble groups with nilpotent length at most r, where r is a positive integer. ~: the class of all finite soluble =-groups, where = is a set of primes. ~: the class of all finite p-groups. ~: ~a~. Up: the class of all finite soluble groups with central p-socle. ='= {PIP is a prime, p~=} for each set of primes =. Section 1 A Fitting class ~ is a set of groups with the following closure properties: 1. If Ge~,, and f : N ~ - N f < G , then N E ~ 2. If N, M ~ ~, and N, M < G = N M , then G ~ It is immediate that given a Fitting class ~,, each group G possesses a normal subgroup which is uniquely maximal among the normal subgroups of G which lie in This is the ~-radical of G, denoted by G~. It is easy to show that for any group G and Fitting classes X and Yg, G=~ ~ = G~ ~ G~, and that if N < G, N~ = N c~ G~. If for every group G, Go~ is maximal among the subgroups of G which lie in ~ then ~ is called a normal Fitting class. Lockett defines in 1-10] the Fitting class ~ * . For a given Fitting class ~, W* = {GI(G • G)~ is subdirect in G x G}. Since for two Fitting classes X and ~ , it is true that (X c~~)* = X* c~~*, another Fitting class ~ , may be defined by ~ , = ('] { X [ X is a Fitting class with X * = ~ - * } . It is clear that for a given Fitting class, J~, is uniquely minimal among the Fitting classes "with the same star as W". The following facts about ~ * will be needed. Theorem 1.1 [10, 4]. Let X and r be Fitting classes. 1. I f X ~ ~, then X* ~_~J*, and Y(, ~_~r 2. X * = ~ J * ~ X,~_~/=_X*. 3. For any groups G and H, (G x H)~c, = G~, x H~,. 4. If G~X*, then [G, AutG] <G~,. If a Fitting class satisfies the identity ~- = ~ * , then ~- is called a Lockett class. The set of Fitting classes {WI~,-~ X c r,~*} is known as the Lockett section of denoted here by L(@). Let G and H be groups. Then GZH denotes the regular wreath product of G with H. I f K =<G, then we will denote by/'~ the subgroup of the base group of G%H which is isomorphic to the direct product of [HI copies of K. Thus C denotes the base group of G%H.