Group theory with one more axiom

Question

I am looking for the answer to the following question. How many models (and which ones) have (accurate to isomorphism) the group theory with an additional axiom?

$\forall a, \forall b, \forall c, \forall d, \forall e, \forall f, \left ( a= b \vee a=c \vee a=d \vee a=e \vee a=f \vee b=c \vee b=d \vee b=e \vee b=f \vee c=d \vee c=e \vee c=f \vee d=e \vee d=f \vee e=f\right )$

Answer 1

The axiom is equivalent to saying that there is no injective function from a $6$-element set to the underlying set of the group, which is in turn equivalent to saying that the group has fewer than $6$ elements.

The trivial group has order $1$.

The cyclic group $\mathbb{Z}_{2}$ has order $2$.

The cyclic group $\mathbb{Z}_{3}$ has order $3$.

The cyclic group $\mathbb{Z}_{4}$ and the Klein four group $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ have order $4$.

Finally, the cyclic group $\mathbb{Z}_{5}$ has order $5$.

The total number of models is therefore $6$.

Answer 2

Groups with at most $5$ elements are all abelian.

In cardinal $1, 2, 3$ , there is only one each time : the cyclic one.

In cardinal $4$, you have $\mathbb{Z}/4$ and $(\mathbb{Z}/2)^2$

In cardinal $5$, you have $\mathbb{Z}/5$, which is the only one.