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RESEARCH PRODUCT

OnF-Subnormal Subgroups andF-Residuals of Finite Soluble Groups

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All groups that we consider are finite and soluble. Recall that a formation is a class of groups which is closed under homomorphic images and subdirect products. Hence, if F is a formation and G is a group which is a direct product of the subgroups A and B, then G is in F if and only if A and B lie in F. More generally, Doerk and w x Hawkes 4, IV, 1.18 proved that if G is a group such that G s A = B, then G s A = B , where G is the F-residual of G, that is, the smallest normal subgroup of G with quotient in F. The main purpose of this paper is the development of this result by means of the concept of F-subnormal subgroup. Suppose that F is a saturated formation. A maximal subgroup M of a Ž . group G is called F-normal in G if GrCore M g F. A subgroup H of G G is called F-subnormal in G if either H s G or there exists a chain H s H F H F ??? F H s G such that H is an F-normal maximal 0 1 n i subgroup of H for 0 F i n. It is clear that if F s N, the saturated iq1 formation of all nilpotent groups, the F-subnormal subgroups of G are exactly the subnormal subgroups of G. Let F be a subgroup-closed saturated formation containing N. It is rather easy to see that if F is closed under the product of normal subgroups, then G s A B F for every pair of subnormal subgroups A and B such that G s AB. This result does not remain true if A and B are