# Orthogonal IAPS

From Groupprops

*This article describes a particular IAPS of groups, or family of such IAPSes parametrized by some structure*

## Definition

The **orthogonal IAPS** associated to a commutative unital ring (usually a field) is the IAPS of groups defined as follows:

- Its member is the orthogonal group : the group of matrices such that is the identity matrix
- Its block concatenation map is described as follows:

## Inside the GL IAPS

`Further information: Orthogonal IAPS in GL IAPS`

The orthogonal IAPS is a sub-IAPS of the GL IAPS, which comprises the general linear groups. It is in fact a saturated sub-IAPS, and the quotient space can be identified with diagonalizable bilinear forms.