You are wrong but I think not fundamentally: your argument about the order of the

image is right, but you oversaw the fact that the order of any of the three 2-cycles

you wrote there indeed is 2, so you can define , and

the same can be done with any of the other two elements of order 2 in that direct sum.

Of course, the above isthe whole story since you still havenot

to figure out how to define f on the other elements of that sum...

Tonio