A doubt about Exercise 12 in textbook Algebra by Saunders MacLane and Garrett Birkhoff

Question

I'm doing Exercise 12 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.

For $T \subset G$ and fixed $a \in G$ show that the assignment $g T \mapsto a g T$ is a permutation $h_{a}: G / T \rightarrow G / T$ on the set $G / T$ of right cosets of $T$ in $G$.

IMHO, for the map $g T \mapsto a g T$ to be well-defined, it must be the case that $gT=hT \implies agT = ahT$. This means $gh^{-1} \in T \implies (ag)(ah)^{-1} \in T$, or equivalently $gh^{-1} \in T \implies agh^{-1}a^{-1} \in T$. This means $T$ is a normal subgroup.

Could you please confirm that for the exercise to be true, we need stronger hypothesis, i.e. $T$ is a normal subgroup?

Answer 1

Here is @Arturo Magidin's comment that answers my question. I post it here to remove this question from unanswered list. All credits are given to @Arturo Magidin.

You’ve got the conditions for equality wrong. $gT=hT$ if and only if $g^{-1}h\in T$, not $gh^{-1}\in T$.