Axiomatic Systems, Binary Operations, and Groups

Question

We have the four axioms of the group (1. Closure, 2. Associativity, 3. Neutral Element, 4. Inverse Element). Axiom 5: Supposing we have group $G$ which is the group of the functions $f,g,h,k:{1,2,3,4} \to {1,2,3,4}$. Let function $f$ be the Identity function $(f(1)=1, f(2)=2...)$, Let function $g$ be in such way that, $g(1)=2, g(2)=1, g(3)=3, g(4)=4$ and let function $h$ be in such way that $h(1)=1, h(2)=2, h(3)=4, h(4)=3$. And let function $k$ be in such way that $k=g\circ h$.

The questions:

1. Prove that $G$ with the composition of the functions, is a model to the axioms 1,2,3,4,5.
2. Prove that the system 1,2,3,4,5 is not categorical.

I tried to solve that by myself, and to do that, I have made a table, Which didn't help me by any way, and now after 3 hours of trying to solving it, I am posting it there.

Answer 1

Because function composition is associative, this operation is associative. The neutral element is the identify function $f$. Since all the functions are bijective, they are invertible. All that's left to check is closure.

We can interpret these functions as permutations of the numbers $1,2,3$ and $4$. Since $g$ tranposes $1$ and $2$ and $h$ transposes $3$ and $4$ these functions commute. This also shows that $g,h$ and $k$ are their own inverses.

Answer 2

Here is a Cayley table for your group:

$o\;f\;g\;h\;k$

$f\;f\;g\;h\;k$

$g\;g\;f\;k\;h$

$h\;h\;k\;f\;g$

$k\;k\;h\;g\;f$

In fact, it is isomorphic to the Klein vierergruppe.