Group Structure
The rotation group is a group under function composition (or equivalently the product of linear transformations). It is a subgroup of the general linear group consisting of all invertible linear transformations of Euclidean space.
Furthermore, the rotation group is nonabelian. That is, the order in which rotations are composed makes a difference. For example, a quarter turn around the positive x-axis followed by a quarter turn around the positive y-axis is a different rotation than the one obtained by first rotating around y and then x.
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Read more about this topic: Rotation Group SO(3)
Famous quotes containing the words group and/or structure:
“[The Republicans] offer ... a detailed agenda for national renewal.... [On] reducing illegitimacy ... the state will use ... funds for programs to reduce out-of-wedlock pregnancies, to promote adoption, to establish and operate children’s group homes, to establish and operate residential group homes for unwed mothers, or for any purpose the state deems appropriate. None of the taxpayer funds may be used for abortion services or abortion counseling.”
—Newt Gingrich (b. 1943)
“One theme links together these new proposals for family policy—the idea that the family is exceedingly durable. Changes in structure and function and individual roles are not to be confused with the collapse of the family. Families remain more important in the lives of children than other institutions. Family ties are stronger and more vital than many of us imagine in the perennial atmosphere of crisis surrounding the subject.”
—Joseph Featherstone (20th century)