Special orthogonal similitude group

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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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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.

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