every permutation is either even or odd,but not both How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...
Thanks in advance...

It is enough to show that the product of an odd number of transpositions cannot be the identity.
Every permutation of a finite set $S$ is a unique product of disjoint cycles in which every element of $S$ occurs exactly once (where we include fixed points as 1-cycles).  Let $p$ be any permutation of $S$, let $(ij)$ be a transposition ($i,j \in S$), and let $q=p \cdot (ij)$. It is easy to check that if $i$ and $j$ are in the same cycle in $p$, then that cycle splits into two in $q$; if $i$ and $j$ are in different cycles in $p$, then those cycles merge into one in $q$.  Cycles of $p$ not containing either $i$ or $j$ remain the same in $q$.  Therefore, $q$ has either one more or one less cycle than $p$ does.  
Now let $t$ be any product of an odd number of transpositions.  Then by the above, multiplying any permutation by $t$ changes the parity of the number of cycles in the permutation.  Therefore $t$ cannot be the identity.
