O'Reilly logo

Classical Geometry: Euclidean, Transformational, Inversive, and Projective by G. W. Tokarsky, A. C. F. Liu, J. E. Lewis, I. E. Leonard

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

CHAPTER 10

SYMMETRY AND GROUPS

10.1 More About Groups

Recall that a set together with a binary operation · is called a group if the following conditions are satisfied:

1. is closed under the binary operation; that is, if x and y are elements of , then so is x · y.
2. The associative law holds. If x, y, and z are elements of , then

equation

3. There is an identity element e in . For every x in , e · x = x · e = x.
4. is closed with respect to inversion. For every member x in , there is another member x′ also in such that x · x′ = x′ · x = e.

When dealing with groups in general, the binary operation ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required