Exercise
Only one part of the proposition is not valid if the condition that N is a normal subgroup is relaxed to N just being a subgroup. Which one?
Part 1
Part 2
Part 3
Part 4
No.
Only associativity is used in the proof.
No.
Only subgroup properties are used in the proof.
Yes. Take a = 1; a little argument shows that this condition is equivalent to
N being a normal subgroup of G.
No. The proof of this assertion uses left coset properties only.