In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group.
In this section, let G be a finite p-group of order |G| = pn and exponent exp(G) = pk.
Groups in which Ω(G) ≤ Z(G) were studied by John G. Thompson and have seen several more recent applications.
The dual notion, groups with [G,G] ≤ ℧(G) are called powerful p-groups and were introduced by Avinoam Mann.
These groups were critical for the proof of the coclass conjectures which introduced an important way to understand the structure and classification of finite p-groups.