Thread: Rigid Motions

because M/SM is isomorphic to the 2-element group {-1,1}.

and these are the cosets:

SM = rotations, SM*R = reflections.

this is the isomorphism:

SM<--->1
SM*R <---> -1

note that det(reflection*reflection) = det(rotation) = 1 = (-1)(-1) = det(reflection)det(reflection)
det(rotation*reflection) = det(reflection) = -1 = (1)(-1) = det(rotation)det(reflection)
det(reflection*rotation) = det(reflection) = -1 = (-1)(1) = det(reflection)det(rotation)
det(rotation*rotation) = det(rotation) = 1 = (1)(1) = det(rotation)det(rotation)

that is, the homomorphism det: GL(2,R) ---> R given by det:A--->det(A) yields a surjective homomorphism (which we also call det)

det:M--->{-1,1} (which sends A-->det(A)), which yields an ISOmorphism from M/SM---->{-1,1},

the cosets of SM in M are all the elements of M that all have the same determinant, we have only 2, the index of SM in M is 2.

one of these sets is SM, the rotation subgroup of M, the other is SM*R (where R is any representative reflection), the set of "improper rotations"

or reflections (these are called reflections because they are orientation-reversing, and any reflection is of order 2:

R*R = I, the 2x2 identity matrix).

That was a huge help. Thank you.