# 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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## Definition

Let $k$ be a field and $n$ be a natural number. The special orthogonal similitude group of order $n$ over $k$ is defined as the group of matrices $A$ such that $AA^t$ is a scalar matrix whose scalar value is a $n^{th}$ 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 $n$. Then, the map sending $k$ 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.