Thompson subgroup

In mathematical finite group theory, the Thompson subgroup ${\displaystyle J(P)}$ of a finite p-group P refers to one of several characteristic subgroups of P. John G. Thompson (1964) originally defined ${\displaystyle J(P)}$ to be the subgroup generated by the abelian subgroups of P of maximal rank. More often the Thompson subgroup ${\displaystyle J(P)}$ is defined to be the subgroup generated by the abelian subgroups of P of maximal order or the subgroup generated by the elementary abelian subgroups of P of maximal rank. In general these three subgroups can be different, though they are all called the Thompson subgroup and denoted by ${\displaystyle J(P)}$.