A note on conjugacy of supplements in soluble periodic linear groups

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 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), no. 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), no. 3, 417-419].