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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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.
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.