Normality theorem for elementary symplectic group with respect to an alternating form

TL;DR

This paper generalizes a normality theorem for elementary symplectic groups, extending previous results from standard to arbitrary invertible skew-symmetric matrices of Pfaffian one, over commutative rings.

Contribution

It extends Kopeiko's normality theorem for symplectic groups to include any invertible skew-symmetric matrix of Pfaffian one, broadening the class of symplectic groups covered.

Findings

Proves normality of elementary symplectic groups with respect to arbitrary invertible skew-symmetric matrices.

Generalizes previous results from standard to more general symplectic forms.

Enhances understanding of the structure of symplectic groups over rings.

Abstract

A.A. Suslin proved a normality theorem for an elementary linear group, which says that an elementary linear group of size bigger than or equal to 3 over a commutative ring with unity is normal in the general linear group of same size. Subsequently, V.I. Kopeiko extended this result of Suslin for a symplectic group defined with respect to the standard skew-symmetric matrix of even size. Here we generalise the result of Kopeiko for a symplectic group defined with respect to any invertible skew-symmetric matrix of even size of Pfaffian one.