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 According to , the ''p''-groups of symplectic type were classified by P. Hall in unpublished lecture notes, who showed that they are all a central product of an extraspecial group with a group that is cyclic, dihedral, quasidihedral, or quaternion.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.

References

* *{{Citation , last1=Thompson , first1=John G. , author1-link=John G. Thompson , title=Nonsolvable finite groups all of whose local subgroups are solvable , url=https://www.ams.org/journals/bull/1968-74-03/S0002-9904-1968-11953-6/home.html , doi=10.1090/S0002-9904-1968-11953-6 , mr=0230809 , year=1968 , journal= Bulletin of the American Mathematical Society , issn=0002-9904 , volume=74 , pages=383–437, doi-access=free Finite groups