Another Cycle Decomposition Question
In let be a cycle of length 4, and be a transposition.
Show is even, and briefly explain why
Let , then the decomposition would be , and let . This means that would be , which is even. Think I've gone about this in the correct way?
For any transposition , , this means that .
For our transpositions above, , so even if , it would still not equal the identity transposition. Would this be enough to briefly explain?
I managed to attempt the first couple of sections of the question, but this last part has got me very confused:
Deduce that is either a cycle of length 3, or a product of 2 disjoint transpositions. Conclude that for some not necessarily disjoint permutations
Would greatly appreciate some help in how to go about this problem.
Thanks in advance