Orthogonal similitude group for the standard dot product

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 $n$ 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 map

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.

Relation with other linear algebraic groups

Subgroups

• 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.
• Orthogonal group: The subgroup comprising matrices with factor of similitude $1$.
• Special orthogonal group: The subgroup comprising matrices with factor of similitude $1$ and determinant $1$.

Particular values

Finite fields

Size of field Order of matrices Common name for the orthogonal similitude group Order of the orthogonal similitude group Comment
2 1 Trivial group $1$ Trivial
3 1 Cyclic group:Z2 $2$ group of prime order
4 1 Cyclic group:Z3 $3$ group of prime order
5 1 Cyclic group:Z4 $4 = 2^2$ cyclic group of prime power order
2 2 Cyclic group:Z2 $2$ group of prime order
3 2 SmallGroup(16,8) $16 = 2^4$ group of prime power order, hence nilpotent
4 2 Alternating group:A4 $12 = 2^2 \cdot 3$ solvable but not nilpotent or supersolvable