One particular permutation $\sigma $ of $5$ elements such that $\sigma ^{-1} (j) \le \sigma (j)$ 
As  for  options  $1.$  and  $2.$  I  have  taken  a  few  examples  randomly and  found  them  to  be  correct  but  could  not  generalize  and I am  clueless  about options $3.$  and  $4.$ The  answer   keys  that  came  with   it  says  all  the  options   are  correct.

For $2$., sum the inequality over all $j$. Then both sides are equal, so equality must hold for all $j$.
$2$. implies $1$. by the definition of $\sigma^{-1}$.
$3$. and $4$. imply each other, since the total number of elements is odd (namely $5$).
For $3$., note that $\sigma(k)=\sigma^{-1}(k)$ implies that the elements that $\sigma$ doesn't leave invariant come in pairs.
