I'm looking at an old group theory question that I botched, and I'm wondering how I should proceed. The question asks to write $(152)(45)$ as a product of powers of $(12)$ and $(12345)$.
The 2-cycle $(45)$ is easily enough written as $(12345)^3(12)(12345)^{-3}$. It's also easy to break down the 3-cycle $(152)$ into $(12)(15)$, which partly satisfies the requirements. I just need to figure out how to express $(15)$ as a product of exponents of $(12)$ and $(12345)$. This is where I'm a bit stuck.
I thought about expressing $(15)$ as $(12345)^j(12)^k(12345)^{-m}$ for positive integers $j$, $k$ and $m$, but I couldn't see if/how that would help. Then I tried $(45)(34)(23)(12)$ (knowing that I could decompose each of those transpositions as I did above) but it doesn't produce a 2-cycle. Is there another way to make $(15)$ that I'm missing?
Thanks.