In Needham's Visual Complex Analysis on pdf page 57, book page 37, Needham says that the set of indirect/opposite motions (motions reflecting the angle of a vector input) does not form a subgroup of motions (with the binary operation being functional composition).
I can't seem to figure out why it does not form a group... the inverse of a map reversing an angle is itself, thus the identity of the group is the identity map, and composition is trivially associative. Where am I going wrong?
Groups must be closed under their multiplication. In this case, the multiplication is function composition. The set of opposite motions is the set of function that reverse the angle. If you reverse the angle twice, you get a direct motion which is not an opposite motion. Therefore the composition of any two opposite motions is not an opposite motion and thus composition on this set is not closed and the set is not a group. To further emphasize this, the identity function is not an opposite motion.