Locally nilpotent injectors of CC-groups. (English) Zbl 1293.20037
Martin-Peinador, Elena (ed.) et al., Contribuciones matemáticas en homenaje al profesor D. Antonio Plans Sanz de Bremond. Zaragoza: Secretariado de Publicaciones, Universidad de Zaragoza (ISBN 84-7733-158-8/pbk). 233-238 (1990).
From the text: The aim of this note is to study the characterization of the conjugacy of the injectors in the locally nilpotent case, a question which is similar to those raised and solved by the authors [in Proc. Am. Math. Soc. 106, No. 3, 605-610 (1989; Zbl 0672.20017)]. Thus, from now, injector will mean locally nilpotent injector and the main result in this paper is the following.
Theorem A. For a locally soluble CC-group \(G\) the following assertions are equivalent: (1) The injectors of \(G\) are conjugate. (2) \(G\) has only finitely many injectors. (3) If \(V\) is an injector of \(G\) then \(V^G/V_G\) is finite. (4) \(G\) has a locally nilpotent-by-finite subgroup \(L\) containing all the injectors of \(G\).
For the entire collection see [Zbl 1271.57001].
Theorem A. For a locally soluble CC-group \(G\) the following assertions are equivalent: (1) The injectors of \(G\) are conjugate. (2) \(G\) has only finitely many injectors. (3) If \(V\) is an injector of \(G\) then \(V^G/V_G\) is finite. (4) \(G\) has a locally nilpotent-by-finite subgroup \(L\) containing all the injectors of \(G\).
For the entire collection see [Zbl 1271.57001].
MSC:
| 20F17 | Formations of groups, Fitting classes |
| 20F19 | Generalizations of solvable and nilpotent groups |
| 20E07 | Subgroup theorems; subgroup growth |
| 20E25 | Local properties of groups |
| 20F22 | Other classes of groups defined by subgroup chains |
| 20F24 | FC-groups and their generalizations |
