On a theorem of Ito and Szép
AbstractA 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\in G such that 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 \langle H,T\rangle. In this paper, we prove that 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.