(1)

(2)Prove that if is a normal subgroup of the finite group and then is the unique subgroup of of order .

Regarding notation, means the number of left cosets of in , that is, the number of elements in the set . Also, is just the order of , that is, the number of elements in . Finally, denotes the greatest common divisor of the integers .Let be a subgroup of some group (possibly infinite) , and let be a subgroup of . Prove that .

I could probably get these with enough time, but I'm fresh out of it. Any hints (or heck, even just complete solutions) would be much appreciated!