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.
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
, then
3. There is an identity element e in . For every x in , e · x = x · e = x.
. 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.
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 ...
Get Classical Geometry: Euclidean, Transformational, Inversive, and Projective now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
