I came across this problem that asks me to convert (12)(34) into a product of 5 cycles. After testing for many different combinations i get (12345)(14352)(12345). The way I do it is this: (12)(23)(345)(345)(345)(512)(512)(512)(23)(34). The middle values are multiplied 3 times because they 3-cycles are of order 3. I first convert them into a product of 3 cycles first, that is what (12)(23)=(123) and (23)(34)=(234) are meant for. Then because I need 4,5 to make the first cycle into a 5 cycles, I multiply by (345). I need 1,5 to make the second cycle into a 5 cycle, so I pick (512). And it happens to give me an answer. However, I am wondering if this method is the right method. Because I am wondering what if the middle (345)(345)(512)(512) does not give me a 5 cycle.
Hence, I am wodering if there is an effective way to convert a product of 2-cycles into a product of 5-cycles?