Suppose there is a group of permutation of a set with
Now this group has identity permutation( ) and inverse permutation( ) namely: and
But there is a theorem which says that:
"No matter how is written as a product of transpositions, the number of transpositions is even."
Now by decomposing to cycle I get a transposition like this:
It consists of a single transposition. So the number of transposition for this identity permutation is odd.
Can anyone kindly tell me why this example of transposition for identity permutation does not violate the above mentioned theorem?
And why is my logic wrong here and where am I wrong?