Let $\displaystyle G$ be the subset of $\displaystyle S_4$ consisting of the permutations:

$\displaystyle \varepsilon = \begin{pmatrix} 1 & 2 & 3 & 4\\ 1 & 2 & 3 & 4 \end{pmatrix}$

$\displaystyle f = \begin{pmatrix} 1 & 2 & 3 & 4\\ 2 & 1 & 4 & 3 \end{pmatrix}$

$\displaystyle g = \begin{pmatrix} 1 & 2 & 3 & 4\\ 3 & 4 & 1 & 2 \end{pmatrix}$

$\displaystyle h = \begin{pmatrix} 1 & 2 & 3 & 4\\ 4 & 3 & 2 & 1 \end{pmatrix}$

Now show that $\displaystyle G$ is a group of permutations.

Now the problem is how do I prove this? Because I know that in order to be a group of permutations I've to show the following $\displaystyle 4$ properties:

a) The operation $\displaystyle \circ$ of composition of functions qualifies as an operation on the set of all the permutations of $\displaystyle A$

b) This operation is associative.

c) There is a permutation $\displaystyle \varepsilon$ such that $\displaystyle \varepsilon \circ f = f$ and $\displaystyle f \circ \varepsilon = f$ for any permutation $\displaystyle f$ of $\displaystyle A$.

d) Finally for every permutation $\displaystyle f$ of $\displaystyle A$ there is another permutation $\displaystyle f^{-1}$ of $\displaystyle A$ such that $\displaystyle f \circ f^{-1} = \varepsilon$

and $\displaystyle f^{-1} \circ f = \varepsilon$

Do I exhaust all the possible values and show that the above four properties hold true for all combinations?

Is there any easy way to prove that the set of all the above permutations is a group?