An extraspecial \(p\)-group is one of the closest nonabelian analogues of an elementary abelian group. All noncommutativity is concentrated in a center of order \(p\), while the central quotient is a symplectic vector space. [@isaacs2008]
Definition
An extraspecial group has order \(p^{1+2n}\). The quotient \(V=P/Z(P)\) is a \(2n\)-dimensional vector space over \(\mathbf F_p\).
Core result
Class at most \(2\) makes the commutator linear in each variable. If \(xZ(P)\) is orthogonal to every vector, then \(x\) centralizes all of \(P\) and lies in \(Z(P)\), proving nondegeneracy. The full classification also uses exponent or quadratic data; see Section 4D.
Structural properties
- The order is \(p^{1+2n}\) and the central quotient is elementary abelian.
- Maximum abelian subgroups correspond to maximal totally isotropic subspaces and usually have order \(p^{n+1}\).
- For odd \(p\), two central-product types occur at each order, distinguished by exponent or quadratic data.
- For \(p=2\), \(D_8\) and \(Q_8\) are the two groups of order \(8\), and central products produce the higher types.
- Automorphisms induce symplectic or related orthogonal transformations on \(P/Z(P)\).
This linearization converts local \(p\)-group questions into finite symplectic geometry and recurs in Thompson replacement and local simple-group theory.
Example and boundary
Knowledge network
The finite p-group core article supplies center and Frattini theory, Thompson subgroups use maximum abelian subgroups, and symplectic groups describe induced automorphisms.
Proof and sources
Definition, class-two commutators, and central products are in Section 4D; maximum-abelian applications continue in Chapter 7. [@isaacs2008]
