# Orthogonal IAPS

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 $R$ (usually a field) is the IAPS of groups defined as follows:

• Its $n^{th}$ member is the orthogonal group $O(n,R)$: the group of $n \times n$ matrices $A$ such that $AA^t$ is the identity matrix
• Its block concatenation map $\Phi_{m,n}$ is described as follows:

$\Phi_{m,n}(A,B) = \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}$

## 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.