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
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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.