Semipermutable subgroups and s-semipermutable subgroups in finite groups

Suppose that H is a subgroup of a finite group G. We call H is semipermutable in G if HK = KH for any subgroup K of G such that ( divide H divide , divide K divide ) = 1; H is s-semipermutable in G if HG(p) = G(p)H, for any Sylow p-subgroup G(p) of G such that ( divide H divide , p) = 1. These two concepts have been received the attention of many scholars in group theory since they were introduced by Professor Zhongmu Chen in 1987. In recent decades, there are a lot of papers published via the application of these concepts. Here we summarize the results in this area and gives some thoughts in the research process.