Q: Let $f$ and $g$ be permutations on the set $\{1,2,3,4,4,5,6,7\}$, defined as follows $$f=(436)(5124)$$ $$g=(2746135)$$
Write the permutation $fg$ as a product of disjoint cycles, separated by commas, (e.g., $(1,2),(3,4,5),\dots)$. Do not include 1-cycles in your answer.
My thoughts + working: First, I want to combine $f=(436)(5124)$ into a single cycle $(124365)$ (from multiplying left to right using $kp=k\cdot p=p\circ k$) because they are not disjoint because $4$ appears in both cycles.
Thus, $fg=(124365)(2746135)=(17453)(26)$. Is my working correct, and could I have done anything better? (i.e. have I done any unnecessary steps?)
You do not need to rewrite $f$ into a product of disjoint cycles first. Just do the calculation directly with all three cycles!
Looking at $$(436)(5124)(2746135),$$ we have $1\mapsto1\mapsto2\mapsto 7$. So we start $(17\cdots$.
Then $7\mapsto7\mapsto7\mapsto4$, so we continue $(174\cdots$
Then $4\mapsto3\mapsto3\mapsto5$, so now we have $(1745\cdots$
Etc. Continuing this way, we end up with $$(17453)(26)$$ in a single computation.