Special orthogonal similitude group

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 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.

Subgroups

 * subgroup::Special orthogonal group

Supergroups

 * supergroup::Orthogonal similitude group
 * supergroup::Special linear group