theorem 1.31 The set of all rotations about the origin and reflections in lines through the origin is a group called orthogonal group and is denoted by 0(2). SO(2) is a subgroup of index 2 in O(2)....

Extracted text: theorem 1.31 The set of all rotations about the origin and reflections in lines through the origin is a group called orthogonal group and is denoted by 0(2). SO(2) is a subgroup of index 2 in O(2). Det'n: Let P be any point in E². The set P of all lines that passes through P is called pencil of lines through P. We denote REF(P) : REF(P) the smallest group of isometries containing all 24, where l e P. We denote ROT(P) the set of all rotations about P.