$ G = D_6$ (dihedral of order 12, with generator $a$, $b$ where $a^6 = b^2 = e$ and $ba=a^{6−1}b$
$H=⟨a^1b^0⟩ $
Given $(x,y)$ pairs, $(a,b^0), (b,a^5b), (b^0a^1, a^4)$, determine if $xH = yH$.
The exercise is simply to give a yes or no answer to that question.
Stuck on how to answer this, partially because I'm not sure how to compute $H=⟨a^1b^0⟩$. $H=\{a^1b^0, a^1b^0 \cdot a^1b^0, a^1b^0 \cdot a^1b^0 \dot a^1b^0, ... \} =\ ?$ I'm assuming I'm to use the above facts but I'm failing to work out the algebra.