Being isometries, real orthogonal transforms preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between congruence and similarity, as exemplified by SSS (side-side-side) congruence of triangles and AAA (angle-angle-angle) similarity of triangles. The group of conformal linear maps of is denoted for the conformal orthogonal group', and consists of the product of the orthogonal group with the group of dilations. If is odd, these two subgroups do not intersect, and they are a direct product: , where } is the real multiplicative group, while if is even, these subgroups intersect in , so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: .
Being isometries, real orthogonal transforms preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between congruence and similarity, as exemplified by SSS (side-side-side) congruence of triangles and AAA (angle-angle-angle) similarity of triangles The group of conformal linear maps of is denoted for the conformal orthogonal group', and consists of the product of the orthogonal group with the group of dilations. If is odd, these two subgroups do not intersect, and they are a direct product: , where } is the real multiplicative group, while if is even, these subgroups intersect in , so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: .