# Find $3$ non-isomorphic groups of order $2012$

Find $$3$$ non-isomorphic groups of order $$2012$$.

Is the following correct?

First of all, we have the two non-isomorphic abelian groups $$\mathbb Z_{2012}$$ and $$\mathbb Z_{2}\times\mathbb Z_{1006}$$.

Then I want to construct a non-abelian group from $$\mathbb Z_4$$ and $$\mathbb Z_{503}$$. We have $$\text{ Aut }(\mathbb Z_{503})\cong\mathbb Z_{503}^\times\cong\mathbb Z_{502}$$. Let $$a$$ be an element of order $$2$$ in that group (exists because of Cauchy's theorem), then $$\varphi:\mathbb Z_4\to\text{ Aut }(\mathbb Z_{503}), \bar 1\mapsto a$$ is a non-trivial group homomorphism and thus $$\mathbb Z_4\rtimes_\varphi\mathbb Z_{503}$$ is non-abelian.

You could do it even easier. $$2012$$ is an even number so there is also the dihedral group $$D_{1006}$$.
• What you did looks fine. A semidirect product can't be abelian unless it comes from a trivial homomorphism. (which means it is actually a direct product). I just wanted to point on the fact that it is actually very easy to find a non abelian group of even order (if the order is at least $6$) without using semidirect products.