Affiliation:
1. Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” , 9307 Università degli Studi di Napoli Federico II , Complesso Universitario Monte S. Angelo, Via Cintia , Napoli , Italy
Abstract
Abstract
The aim of this short note is to prove that if G is a (homomorphic images of a) soluble periodic linear group and N is a locally nilpotent normal subgroup of G such that N and
G
/
N
{G/N}
have no isomorphic G-chief factors, then two supplements to N in G are conjugate provided that they have the same intersection with N. This result follows from well-known theorems in the theory of Schunck classes (see [A. Ballester-Bolinches and L. M. Ezquerro,
On conjugacy of supplements of normal subgroups of finite groups,
Bull. Aust. Math. Soc. 89 2014, 2, 293–299]), and it appeared as the main theorem of [C. Parker and P. Rowley,
A note on conjugacy of supplements in finite soluble groups,
Bull. Lond. Math. Soc. 42 2010, 3, 417–419].
Reference10 articles.
1. A. O. Asar,
A conjugacy theorem for locally finite groups,
J. Lond. Math. Soc. (2) 6 (1973), 358–360.
2. A. Ballester-Bolinches and L. M. Ezquerro,
On conjugacy of supplements of normal subgroups of finite groups,
Bull. Aust. Math. Soc. 89 (2014), no. 2, 293–299.
3. A. Ballester-Bolinches, S. F. Kamornikov and V. Pérez-Calabuig,
On complements of
𝔉
\mathfrak{F}
-residuals of finite groups,
Comm. Algebra 45 (2017), no. 2, 878–882.
4. M. Curzio,
Some Problems of Sylow Type in Locally Finite Groups,
Math. Methods 5,
Academic Press, Cambridge, 1979.
5. J. D. Dixon,
Complements of normal subgroups in infinite groups,
Proc. Lond. Math. Soc. (3) 17 (1967), 431–446.