A regular \(p\)-group, in Philip Hall's sense, has power behavior close to that of an abelian group. Its collection formulas make element orders, power subgroups, and transfer easier to control. [@isaacs2008]
Definition
Regular here is unrelated to a regular permutation action; it is a power property of a \(p\)-group.
Core result
Regularity passes to quotients, while \(C_p\wr C_p\) violates the regular power formula and contains the critical element of order \(p^2\). Excluding that quotient allows Yoshida's theorem to apply.
Structural properties
- Every abelian \(p\)-group is regular.
- A finite \(p\)-group of nilpotency class less than \(p\) is regular.
- Regularity behaves well under subgroups and quotients.
- Sets generated by elements of bounded order often become subgroups in a regular group.
- At \(p=2\), the class of regular groups is very restricted.
Regularity pushes nonabelian power errors into powers of higher commutators, preserving many abelian-style counting laws.
Example and boundary
Knowledge network
Yoshida explains the transfer consequence, the wreath product is the obstruction, and powerful p-groups are another class with good power behavior.
Proof and sources
Regularity and its Hall–Wielandt/Yoshida role are discussed after Theorem 10.2 and in the Section 10A problems. [@isaacs2008]
