I'm struggling with the idea of expressing permutations as a product of transpositions. For example, a question I've been given is: Let $\alpha, \beta$ be permutations in $S_8$ where
$\alpha=\begin{bmatrix}1&2&3&4&5&6&7&8\\2&3&1&4&7&6&5&8\end{bmatrix}$
$\beta=\begin{bmatrix}1&2&3&4&5&6&7&8\\4&5&6&7&8&1&2&3\end{bmatrix}$
Find $\alpha\circ\beta$ and $\beta\circ\alpha$.
For $\alpha\circ\beta$ I'm getting $(1458)(2736)$, but the solution provided is $(14)(27)(36)(58)$. Similarly, for the second part I got $(1526)(3478)$, but the solution provided is $(15)(26)(34)(78)$.
I guess my question is: Are my solutions correct? If so, how can they be expressed in the form of a product of transpositions like the solutions provided? If my solutions are incorrect, where am I going wrong?
You're right.
One can see this by checking what the individual elements map to: for example,
$$4\stackrel{\beta}{\mapsto} 7\stackrel{\alpha}{\mapsto} 5,$$ so $(\alpha\circ\beta)4=5$, which agrees with your answer, not the supposed solution.