Zhenfeng Wu, Wenbin Guo
October 24, 2020

### Abstract

A subgroup H of a group G is said to be conditionally permutable (or c -permutable for short) in G if, for every subgroup T of G , there exists an element x∈G{x\in G} such that HTx=TxH{HT^{x}=T^{x}H}. A subgroup H of a group G is said to be completely c -permutable in G if, for every subgroup T of G , the subgroups H and T are c -permutable in 〈H,T〉{\langle H,T\rangle}. In this paper, we prove that H/HG{H/H_{G}} is nilpotent if H is a completely c -permutable subnormal subgroup of G . This result generalizes a well-known theorem of Ito and Szép, and gives a positive answer to an open problem in [W. Guo, Structure Theory for Canonical Classes of Finite Groups, Springer, Heidelberg, 2015]. We also use complete c -permutability to determine the p -supersolubility of a group G .