Local definitions of formations of finite groups

A problem of constructing local definitions for formations of finite groups is discussed in the article. The author analyzes relations between local definitions of various types. A new proof of the existence of an $\omega$-composition satellite of an $\omega$-solubly saturated formation is obtained. It is proved that if a nonempty formation of finite groups is $\mathfrak X$-local by Förster, then it has an $\mathfrak X$-composition satellite.