Let be a homomorphism between two groups and . Show that the image of , is a subgroup of ; show also that the kernel of , is a normal subgroup of .
Show that is isomorphic to .
Let be the group of real orthogonal matrices and let be the set of orthogonal matrices with determinant 1 . Show that is a normal subgroup of and that is isomorphic to the cyclic group of order
Give an example of a homomorphism from to with kernel of order