On the Hirsch-Plotkin radical of a factorized group

Abstract

Let the group G = AB be the product of two subgroups A and B. A normal subgroup K of G is said to be factorized if K = (A ∩ K)(B ∩ K) and A ∩ B ≤ K, and this is well-known to be equivalent to the fact that K = AK ∩ BK (see [1]). Easy examples show that normal subgroups of a product of two groups need not, in general, be factorized. Therefore the determination of certain special factorized subgroups is of relevant interest in the investigation concerning the structure of a factorized group. In this direction E. Pennington [5] proved that the Fitting subgroup of a finite product of two nilpotent groups is factorized. This result was extended to infinite groups by B. Amberg and theauthors, who provedin [2] that if the soluble group G = AB with finite abelian section rank isthe product of two locally nilpotent subgroups A and B, then the Hirsch-Plotkin radical (i.e. the maximum locally nilpotent normal subgroup) of G is factorized. If G is a soluble ℒI group and the factors A and B are nilpotent, it was shown in [3] that also the Fitting subgroup of G is factorized. However, Pennington's theorem becomes false for finite soluble groups which are the productof two arbitrary subgroups. For instance, the symmetric group of degree 4 is the product of a subgroup isomorphic with the symmetric group of degree 3 and a cyclic subgroup of order 4, but its Fitting subgroup is not factorized.