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According to Thompson (1968, p.386), the p-groups of symplectic type were classified by P. Hall in unpublished lecture notes, who showed that they are all a central product of an extraspecial group with a group that is cyclic, dihedral, quasidihedral, or quaternion. Gorenstein (1980, 5.4.9) 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 p¹⁺²ⁿ 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

• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 81b:20002 
• Thompson, John G. (1968), "Nonsolvable finite groups all of whose local subgroups are solvable", Bulletin of the American Mathematical Society, 74: 383–437, doi:10.1090/S0002-9904-1968-11953-6, ISSN 0002-9904, MR 0230809