Editing P-group (section)

== Application to structure of a group ==
''p''-groups are fundamental tools in understanding the structure of groups and in the [[classification of finite simple groups]]. ''p''-groups arise both as subgroups and as quotient groups. As subgroups, for a given prime ''p'' one has the Sylow ''p''-subgroups ''P'' (largest ''p''-subgroup not unique but all conjugate) and the [[p-core|''p''-core]] <math>O_p(G)</math> (the unique largest ''normal'' ''p''-subgroup), and various others. As quotients, the largest ''p''-group quotient is the quotient of ''G'' by the [[p-residual subgroup|''p''-residual subgroup]] <math>O^p(G).</math> These groups are related (for different primes), possess important properties such as the [[focal subgroup theorem]], and allow one to determine many aspects of the structure of the group.

=== Local control ===
Much of the structure of a finite group is carried in the structure of its so-called '''local subgroups''', the [[normalizer]]s of non-identity ''p''-subgroups.<ref>{{harv|Glauberman|1971}}</ref>

The large [[elementary abelian group|elementary abelian subgroup]]s of a finite group exert control over the group that was used in the proof of the [[Feit–Thompson theorem]]. Certain [[Group extension#Central extension|central extension]]s of elementary abelian groups called [[extraspecial group]]s help describe the structure of groups as acting on [[symplectic vector space]]s.

[[Richard Brauer]] classified all groups whose Sylow 2-subgroups are the direct product of two cyclic groups of order 4, and [[John Walter (mathematician)|John Walter]], [[Daniel Gorenstein]], [[Helmut Bender]], [[Michio Suzuki (mathematician)|Michio Suzuki]], [[George Glauberman]], and others classified those simple groups whose Sylow 2-subgroups were abelian, dihedral, semidihedral, or quaternion.