Question

Prove that the product of two left cosets of a subgroup N in a group G is another left coset of N in G

Answer 1

Answer:

Let H be a subgroup of a multiplicative group G such that the product of any two of left cosets of H is a left coset of H. Is H normal in G? Prove or disprove.

I got a hint to solve the problem as follows: Let a,b,c∈G then by the given condition, we have

(aH)(bH)=cH⟹a(Hb)H=cH

⟹Hb=a−1cH⟹b∈a−1cH⟹bH=a−1cH=Hb

I am unable to understand the entire second line.

Problem 1: How to get a(Hb)H=cH⟹Hb=a−1cH

Problem 2: Hb=a−1cH⟹b∈a−1cH (Is it due to the fact that as e∈H then eb∈Hb and consequently eb=b∈a−1cH)

If it is correct, please help me to understand the solution.

Another hint is given as (aH)(bH)=cH⟹cH=abH. Using it how to solve the problem.

In the case of 2nd Hint, clearly, e∈H⟹ae∈aH & be∈bH⟹aebe∈aHbH i.e. ab∈cH but abe∈abH⟹ab∈abH i.e ab is common in aHbH and abH then aHbH=abH. (As we know that any two left cosets are either equal of disjoint.)

But how to conclude the proof.