Number of non isomorphic groups of order p(p prime)

Question

What is the number of non isomorphic groups of order p?

I tried searching through various textbooks but couldn't find the solution. Can we use Euler-phi function to calculate the number somehow?

Answer 1

It's one. There's only one group of prime order $p$, the cyclic one.

Answer 2

Take group $G$ of order $p$, where $p$ is prime. Only divisors of $p$ are 1 and $p$ itself. By Lagrange Theorem, any subgroup of $G$ has to have either one or $p$ elements. So the only subgroups of $G$ are the one-element subgroup - containing only neutral element e and the whole $G.$

So take $a\in G$, that is not equal to the neutral element. Then subgroup $\langle a\rangle_G$ generated by $a$ has to be the whole $G$. So I can define group isomorphism $\phi: G \to {\Bbb Z}_p$ by sending $\phi(a)=1$.

Hence all groups of order $p$ are isomorphic to (additive) group $\Bbb Z_p$. So there is only one group of order $p$ up to isomorphism.