Orthogonal similitude group for the standard dot product

Definition
Let $$k$$ be a field and $$n$$ a natural number. The orthogonal similitude group for the standard dot product  of degree $$n$$ over $$k$$, sometimes denoted $$GO(n,k)$$ or $$GO_n(k)$$, is the multiplicative group of all matrices $$A$$ such that $$AA^t$$ is a nonzero scalar matrix. The scalar value is termed the factor of similitude or ratio of similitude of the particular matrix.

This is a special case of the more general notion of orthogonal similitude group for a symmetric bilinear form.

As a functor from fields to groups
Fix $$n$$. Then the map sending a field $$k$$ to the group $$GO(n,k)$$ is a functor.

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

Supergroups

 * Supergroup::Affine orthogonal similitude group

Subgroups

 * Subgroup::Special orthogonal similitude group: This is its intersection with the special linear group. Note that for the factor of similitude for a special orthogonal similitude matrix must be an $$n^{th}$$ root of unity.
 * Subgroup::Orthogonal group: The subgroup comprising matrices with factor of similitude $$1$$.
 * Subgroup::Special orthogonal group: The subgroup comprising matrices with factor of similitude $$1$$ and determinant $$1$$.

Group and subgroup operations

 * intersection with special linear group equals special orthogonal similitude group.