Definition:

$A_5$ is the alternating group on the set {1, 2, 3, 4, 5} using an even numbered amount of swaps of two elements in composition.

Proof:

Associativity: $A_5$ is a subgorup of the integers so since we know that associativity holds for the integers, we also know that associtavity holds for $A_5$

Identity:

$\begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 1 & 2 & 3 & 4 & 5 \end{pmatrix}$

This is the identity. We know this exists because the identity involves zero swaps, which is an even number.

So, the identity exists in $A_5$.

Inverses: Each inverse is done by repeating the same swap, so if the "swap" is included in the group, the inverse must be also.

Ex: If you swap (1, 2), the matrix you get is

$\begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 1 & 3 & 4 & 5 \end{pmatrix}$

and then to take the inverse of that matrix, you just swap (1, 2) again to get

$\begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 1 & 2 & 3 & 4 & 5 \end{pmatrix}$

Similarly, if you swap (1,2) and (3,4), to get the inverse you then have to compose those swaps with (3, 4) and (1, 2). So, ( (1, 2)$\circ$(3, 4) ) $\circ$ ( (3, 4)$\circ$(1, 2) ) gets you back to

$\begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 1 & 2 & 3 & 4 & 5 \end{pmatrix}$

So, all inverses exist in the group.

Subgroups: Some subgroups of $A_5$ are: $A_5$ $A_4$, $A_3$, $A_2$

$A_5$ - groups are always subgroups of themselves.

$A_4$ is the alternating group on the set {1, 2, 3, 4} where there will not always have to be a fixed number.

$A_3$ is the alternating group on the set {1, 2, 3} where exactly one number will always be fixed.

$A_2$ is the alternating group on the set {1, 2} which will always be the identity.