you should tell us what "M" is, too. we don't have your text available for ready reference.

my assumption is that M is the set of all rigid motions (isometries) of the euclidean plane. how many orientations can the plane have?

see if you can find a surjection from M to {-1,1} (under the operation of multiplication for the latter set).

is this surjection a group homomorphism? if so, what is its kernel?