# Is there a non-abelian group of order 49?

Is there any non-abelian group of order $n=49$? I assume there should be at least one but I cannot find an example.

• Actually any group of order $p^2$ for $p$ a prime is abelian. This is a nice exercise using the class equation. – Qiaochu Yuan Feb 25 '12 at 0:17
• $49 = 7^2$. For a given prime $p$ there are only two groups of order $p^2$, and they are both abelian. You should try proving this! – Dylan Moreland Feb 25 '12 at 0:18
• I took the liberty of removing the boldface text... that seems completely unnecessary. – Tyler Feb 25 '12 at 2:39

The only groups of order $49$ are $\mathbb Z_{49}$ and $\mathbb Z_7\times \mathbb Z_7$ since $49$ is a square of a prime number.