PU.M.A. Vol. 18 (2007), No. 3–4, pp. 257–264 Groups with finitely many normalizers of subgroups with intransitive normality relation Fausto De Mari Dipartimento di Matematica e Applicazioni Università di Napoli Federico II Complesso Universitario Monte S. Angelo Via Cintia I - 80126 Napoli (Italy) e-mail: fausto.demari@dma.unina.it and Francesco de Giovanni Dipartimento di Matematica e Applicazioni Università di Napoli Federico II Complesso Universitario Monte S. Angelo Via Cintia I - 80126 Napoli (Italy) e-mail: degiovan@unina.it (Received: February 15, 2007 and in revised form: June 13, 2007) Abstract. A group G is called a T -group if all its subnormal subgroups are normal. In this paper groups are investigated with finitely many normalizers of (infinite) subgroups which do not have the property T . Mathematics Subject Classifications (2000). 20E15 1 Introduction In a famous paper of 1955, B.H. Neumann [7] proved that each subgroup of a group G has finitely many conjugates if and only if the centre Z (G) has finite index, and the same conclusion holds if the restriction is imposed only to conjugacy classes of abelian subgroups (see [3]). Thus central-by-finite groups are precisely those groups in which normalizers of (abelian) subgroups have finite index, and this result suggests that the behaviour of normalizers has a strong influence on the structure of a group. In fact, Y.D. Polovicki˘ ı [8] has shown that if an FC-group G has finitely many normalizers of infinite abelian subgroups, then the factor group G/Z (G) is finite (recall that G is an FC-group if it has finite conjugacy classes of elements). Since it is very easy to prove that any group with finitely many normalizers of cyclic subgroups has the property FC, it follows from Polovicki˘ ı’s theorem that a group has finitely many normalizers of abelian subgroups if and only if it is central-by-finite. Moreover, it has recently been proved that any (generalized) soluble group with finitely many normalizers of non-abelian subgroups has finite commutator subgroup (see [2]). A group G is said to be a T -group if all its subnormal subgroups are normal, i.e. if normality in G is a transitive relation. The structure of finite soluble T - groups was described by W. Gaschütz [5], while D.J.S. Robinson [9] investigated 257