Special orthogonal similitude group

This term associates to every field, a corresponding group property. In other words, given a field, every group either has the property with respect to that field or does not have the property with respect to that field

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

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 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.

Related groups

Subgroups

• Special orthogonal group

Supergroups

• Orthogonal similitude group
• Special linear group