The operation rule of the semidirect product $C_m\rtimes C_k$

Question

It is known that the semidirect product $C_m\rtimes C_k$ is defined by presentation $$ C_m\rtimes C_k=\langle a, b\mid a^m=1, b^k=1, b^{-1}ab=a^e\rangle, $$ where $e^k\equiv1\pmod{m}$. Since every element in $C_m\rtimes C_k$ is of the form $a^{\alpha}b^{\beta}$, where $\alpha\in\{0,1,\ldots,m-1\}$ and $\beta\in\{0,1,\ldots,k-1\}$, there exist integers $x,y$ for which $$ (a^{\alpha}b^{\beta})(a^{\gamma}b^{\delta})=a^{x}b^{y} $$ How to express the integers $x,y$ using $\alpha,\beta,\gamma,\delta$? If the elements where of the form $b^{\beta}a^{\alpha}$ is was easier to use the relation $b^{-1}ab=a^e$, But here I can not see how to use it to "shift" the product to the form $a^xb^y$. Thanks!

Answer 1

Hint. Use the following rule. If $lq\equiv1\pmod m$, then it follows from the equality $b^{-1}ab=a^l$ that $b^{-1}a^qb=(a^l)^q=a$.