Geodesic
On the Hirsch-Plotkin radical of a factorized group
Glasgow mathematical journal, Tome 34 (1992) no. 2, pp. 193-199

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Franciosi, Silvana; Giovanni, Francesco de. On the Hirsch-Plotkin radical of a factorized group. Glasgow mathematical journal, Tome 34 (1992) no. 2, pp. 193-199. doi: 10.1017/S0017089500008715
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