We know by Lagrange's theorem of groups implies that any group of order $p$, where $p$ is prime, is unique up to isomorphism. Moreover, there are non-prime numbers $n$ such that exist only one group of order $n$ up to isomorphism; for example $15$ and $33$. My question is:
Are there conditions for the uniqueness, up to isomorphism, of the groups of a given order?
Edit: By Sylow's theorems it is well known that if $G$ is a group of order $pq$, where $p > q$ are primes, and $q \nmid p-1$ then $G$ is a cyclic group and thus the only group of this order up to isomorphism.
Best Answer
Yes. A natural number $n$ has the property you want if and only if $\gcd(n,\varphi(n))=1$, where $\varphi$ is the familiar totient function of Euler. Such numbers are known as cyclic numbers.