Group of symplectic type

In mathematical finite group theory, a p-group of symplectic type is a "p"-group such that all characteristic abelian subgroups are cyclic.
According to , the "p"-groups of symplectic type were classified by P. Hall in unpublished lecture notes, who showed that they are all a central product of an extraspecial group with a group that is cyclic, dihedral, quasidihedral, or quaternion. gives a proof of this result.
The width "n" of a group "G" of symplectic type is the largest integer "n" such that the group contains an extraspecial subgroup "H" of order "p"1+2"n" such that "G" = "H"."C""G"("H"), or 0 if "G" contains no such subgroup.
Groups of symplectic type appear in centralizers of involutions of groups of GF(2)-type.