Local conjugacy in finite groups (Q1110656)

The subgroups U and V of a finite group G are said to be locally conjugate if the Sylow p-subgroups of U and V are conjugate to one another. The authors consider a group G which contains locally conjugate subgroups U and V and a nilpotent normal subgroup X such that \(G=XU=XV\). If either X is abelian or for each prime p dividing the order of X, the Sylow p-subgroups of G have class at most 2, it is proven that U and V are conjugate. This generalizes a similar result by \textit{G. O. Losey} and \textit{S. E. Stonehewer} for the case that G is solvable [Q. J. Math., Oxf. II. Ser. 30, 183-190 (1979; Zbl 0408.20013)].