Open main menu

Groupprops β

Gillam's abelian-to-normal replacement theorem for metabelian groups

Statement

Suppose p is a prime number and P is a finite p-group. Suppose P is a metabelian group, i.e., its derived length is at most two. Then, the following are true:

  1. For any abelian subgroup of maximum order A in P, there exists an abelian normal subgroup B of P that is contained in the normal closure of A in P and has the same order as A.
  2. For any elementary abelian subgroup of maximum order A in P, there exists a elementary abelian normal subgroup B of P that is contained in the normal closure of A in P and has the same order as A.