The smallest composite integer $n$ such that there is a unique group of order $n$?

Question

We need to find the smallest composite integer $n$ such that there is a unique group of order $n$.

Attempt: Let us suppose that $n= ab$ is a composite integer where $a,b$ are integers such that $a \neq 1 , b \neq 1$.

Then, there are always at least two ways to write $G$ whose order $n$ of the form $ab$. For Ex: $G$ can be either $\mathbb Z_{ab}$ or $ \mathbb Z_a \oplus \mathbb Z_b$

Hence, the uniqueness property upto isomorphism for a composite $n$ should never exist. Where could I have gone wrong?

EDIT: I haven't taken the situation when $a,b$ are relatively prime into account.

Answer 1

The comments are enough to rule out all composites less than $15$. For example, for even composites we can use the dihedral group and the cyclic group.

To show that every group of order $15$ is cyclic, look at the Sylow Theorems article in wikipedia, where the problem is discussed explicitly. Or else look at this old MSE question.

Answer 2

The group of order p^2 where p is prime i.e group of order 4 and 9 can be eliminated from the fact that these are abelian and so isomorphic to either Zp^2 or Zp×Zp The groups of order 6,8,10,12 can be eliminated because they will be isomorphic to dihedral group and cyclic group So the group of order 15 is cyclic,abelian and hence unique