Arch. Math., Voi. 63, i 8 (1994) 0003-889X/94/6301-0001 $ 3.10/0 9 1994 Birkh/iuser Verlag, Basel On finite products of nilpotent groups By BERNHARD AMBERG a n d BURKHARD HOFLING i. Introduetion. A well-known theorem of Kegel [7] and Wielandt [9] states the solubil- ity of every finite group G = AB which is the product of two nilpotent subgroups A and B; see [1], Theorem 2.4.3. In order to determine the structure of these groups it is of interest to know which subgroups of G are conjugate (or at least isomorphic) to a subgroup that inherits the factorization. A subgroup S of the factorized group G = AB is called prefactorized if S = (A c~ S) (B ~ S), it is called factorized if, in addi- tion, S contains the intersection A c~ B. Generally, even characteristic subgroups of G are not prefactorized, as can be seen e.g. from Examples 1 and 2 below. Our first theorem concerns abnormal subgroups. If X is a class of groups, a maximal subgroup S of the finite group G is called X-abnormal if G/So is not an X-group. Further- more, S is called sub-X-abnormal if there is a series of subgroups S=SocS, c... ~S,,_I ~S,=G, such that Si is both maximal and X-abnormal in S~+~. Recall that the characteristic char(X) of a class of groups X is the set of primes p such that X contains a cyclic group of order p. Theorem A. Let the finite group G = AB be the product of two nilpotent subgroups A and B. If X is a class of groups such that char (X) contains the common prime divisors of the orders of A and B, then every sub-X-abnormal subgroup of G has a factorized con- jugate. Since by [3], Proposition V.3.8, ~-normalizers of a finite soluble group G for a saturat- ed formation ~ are in particular sub-j-abnormal, Theorem A implies the following. Corollary 1. Let the finite group G = AB be the product of two nilpotent subgroups A and B. If q~ is a saturated formation such that char(J) contains the common prime divisors of the orders of A and B, then G has a factorized ~-normalizer. In particular G always has a factorized system normalizer. A subgroup S of a group G is pronormal if for every element g !n G the subgroups S and S o are conjugate in (S, Sg). It will be shown in Proposition 1 below that a pronormal subgroup of a finite product of two nilpotent subgroups has at most one prefactorized conjugate. Archiv der Mathematik 63 t