I want to determine what groups $G$ is isomorphic to when the order is prime: say $2,3,$ or $7$.

Question

I think this may be super easy but I am getting too much in my head about it. I know that if $G$ is a positive prime then $G \cong Z/pZ$. Is it as simple as saying $G \cong Z/2Z$ in the case where $|G|=2$. Do I need to show that $G$ is cyclic or abelian or something first. Is that irrelevant? Thanks in advance.

Answer 1

Yes, a group of prime order $p$ is isomorphic to $\mathbb Z/p\mathbb Z$; it is cyclic and Abelian.

In particular, a group of order $2$ is isomorphic to $\mathbb Z/2\mathbb Z$.

$2$ could be replaced with $3$ or $7$ in that statement.

Answer 2

There is only one group of a given prime order: the cyclic one.

This follows from Lagrange's theorem.