Testing whether a binary structure is a group

[Mr Davis 97]

Homework Statement

Consider the binary structure given by multiplication mod 20 on {4, 8, 12, 16}.
Is this a group? If not, why not?

Homework Equations

The Attempt at a Solution

I started by constructing a Cayley table, and working things out. It turns out that 16 acts as an identity element, 4 is the inverse of itself, 12 and 8 are mutual inverses, and 16 is the inverse of itself. One more things to check would be to see if the associative property is satisfied for all elements. However, this would seem to be a very tedious process.

On the other hand, I know that, up to isomorphism, there are only two types of groups of order 4, the cyclic group $\mathbb{Z}_4$ and the Klein four-group. Just by comparing tables, the Cayley table for this binary structure is equivalent to that of the Klein four-group. So is it valid to say, by isomorphism, that this binary structure is also a group, or do I have to explicitly show associativity?

 

[fresh_42]

The isomorphism is sufficient, because you already know, that the other group is associative. Beside that, you only need the closure, i.e. that the multiplication stays inside the set. Associativity is then inherited by $\mathbb{Z}_{20}$ or even by $\mathbb{Z}$ itself.