Split special orthogonal group of degree two

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Definition

Suppose K is a field. The split special orthogonal group of degree two over K is defined as the subgroup of the general linear group of degree two over K given as follows:

\{ A \in GL(2,K) \mid \operatorname{det}(A) = 1, A\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}A^T = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}\}

For characteristic not equal to two, an alternative definition, which gives a conjugate subgroup and hence an isomorphic group, is:

\{ A \in GL(2,K) \mid \operatorname{det}(A) = 1, A\begin{pmatrix}  1 & 0 \\ 0 & -1 \\\end{pmatrix}A^T = \begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix} \}

Over a finite field

For a finite field K, this group is denoted SO(+1,2,K) and is termed the orthogonal group of "+" type. It is also denoted SO(+1,2,q) where q is the size of the field.

It turns out that:

  • If q is a power of 2, i.e., if q is even, then the group is a dihedral group of order 2(q - 1) and degree q - 1.
  • If q is odd, then the group is a cyclic group of order q - 1.

Arithmetic functions

Over a finite field

We consider here the group SO(+1,2,q) = SO(+1,2,K) where K is a field (unique up to isomorphism) of size q.

Function Value Similar groups Explanation
order Case q odd: q - 1
Case q even: 2(q - 1)

Particular cases

q (field size) p (underlying prime, field characteristic) exponent on p giving q special orthogonal group SO(+1,2,q) order of group second part of GAP ID (GAP ID is (order,2nd part) Comments
2 2 1 cyclic group:Z2 2 1
3 3 1 cyclic group:Z2 2 1
4 2 2 symmetric group:S3 6 1
5 5 1 cyclic group:Z4 4 1
7 7 1 cyclic group:Z6 6 2
8 2 3 dihedral group:D14 14 1
9 3 2 cyclic group:Z8 8 1
11 11 1 cyclic group:Z10 10 2
13 13 1 cyclic group:Z12 12 2
16 2 4 dihedral group:D30 30 3
17 17 1 cyclic group:Z16 16 1