# Definition:Special Orthogonal Group

## Definition

Let $k$ be a field.

The special ($n$th) orthogonal group (on $k$), denoted $\map {\mathrm {SO} } {n, k}$, is:

the set of all proper orthogonal order-$n$ square matrices over $k$
under (conventional) matrix multiplication.

That is: $\map {\mathrm {SO} } {n, k} = \map {\mathrm O} {n, k} \cap \SL {n, k}$