Let $G$ be a group of order $5$ with elements $a, b, c, d, 1$ where $1$ is the identity element. This is the definition of the group.
We all know that this can't be a group because any group of order $5$ is abelian but according to my definition this group is not abelian. But my question is why can't this be a group when it satisfies all the criteria mentioned in the definition of a group.
I wish to find a reason for its ability of not being a group just from the definition of group. For example one can say that the diagonal elements are $1$, which means we have subgroups of order $2$, which is not possible for a group of order $5$. But this is not what I am seeking. Please explain me using nothing more than just the four criteria of the definition of group.
Best Answer
You see that it's an operational system, has a neutral element, and is invertible straight from the table, so the only thing left to check is associativity.
Here, we have $(aa)c = 1c = c \neq b = ad = a(ac)$.
(The good news is that, if an operational system isn't associative, it usually doesn't take too long to find a counterexample.)