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 $a, b, c\in G$ then by the given condition, we have
$$(aH)(bH)=cH \implies a(Hb)H=cH$$
$$\implies Hb=a^{-1}cH\implies b\in a^{-1}cH \implies bH=a^{-1}cH=Hb$$
I am unable to understand the entire second line.
Problem 1: How to get $a(Hb)H=cH\implies Hb=a^{-1}cH$
Problem 2: $Hb=a^{-1}cH\implies b\in a^{-1}cH$ (Is it due to the fact that as $e\in H$ then $eb\in Hb$ and consequently $eb=b\in a^{-1}cH$)
If it is correct, please help me to understand the solution.
Another hint is given as $(aH)(bH)=cH \implies cH=abH$. Using it how to solve the problem.
In the case of 2nd Hint, clearly, $e\in H \implies ae\in aH ~~\& ~~be\in bH \implies aebe\in aHbH ~~ i.e. ~~ab \in cH$ but $abe\in abH \implies ab\in abH$ 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.
Best Answer
Edited to make the argument a bit more efficient.
I think this is more or less what the first hint is suggesting:
$aHbH = cH$, so $ab = aebe \in aHbH = cH$, where $e$ is the identity element. Multiplying on the left by $a^{-1}$ gives us $b \in a^{-1}cH$. Thus the left coset containing $b$ is $a^{-1}cH$, and so $bH = a^{-1}cH$.
Now, $aHb = aHbe \subseteq aHbH = cH$, so $Hb \subseteq a^{-1}cH = bH$ (the last equality was proved in the previous paragraph).
We have established that $Hb \subseteq bH$. This holds for any $b$, so in particular it holds for $b^{-1}$, and so $Hb^{-1} \subseteq b^{-1}H$. Multiplying on the left and right by $b$ gives us $bH \subseteq Hb$.
We have shown both containments, so $bH = Hb$, hence $H$ is normal.