Compute, up to isomorphism, all groups of order $21$

Question

I'm trying to solve this problem from my group theory course:

Compute, up to isomorphism, all groups of order $21$

I've first considered the abelian case, whih gives me as Elemental Divisors $3$, $7$, so one group is $$\mathbb{Z}_3\oplus \mathbb{Z}_7=\boxed{\mathbb{Z}_{21}}$$ After this, I don't know what to do. I have trouble with this kind of problems in general, where I'm asked to find all groups of certain order (The abelian case is easy to handle, I have trouble when the group is non-abelian). What is a general tip for solving this kind of problems? What is the solution in my example? Any help will be appreciated, thanks in advance.

Answer 1

I just found my answer:

Given $|G|=21=3\cdot 7$ (product of two primes), and given that $3|(7-1)$, from my textbook I know that the only possible groups (up to isomorphism) are:

• $G\cong\mathbb{Z}_3\oplus\mathbb{Z}_7\cong\mathbb{Z}_{21}$
• $G\cong\mathbb{Z}_7\rtimes\mathbb{Z}_3$