# Special orthogonal similitude group

This article defines a natural number-parametrized system of algebraic matrix groups. In other words, for every field and every natural number, we get a matrix group defined by a system of algebraic equations. The definition may also generalize to arbitrary commutative unital rings, though the default usage of the term is over fields.

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## Contents

## Definition

Let be a field and be a natural number. The **special orthogonal similitude group** of order over is defined as the group of matrices such that is a scalar matrix whose scalar value is a root of unity.

Equivalently, the special orthogonal similitude group is the intersection of the special linear group with the orthogonal similitude group.

## As a map

### As a functor

Fix . Then, the map sending to the special orthogonal similitude group is a functor.

Note that the special orthogonal similitude groups do *not* form a sub-IAPS of the GL IAPS. In other words, a block concatenation of two special orthogonal similitude matrices need not be a special orthogonal similitude matrix. The problem is that the factor of similitude need not be equal for both.