[Math] Abelian groups of order $14,27,30,$ and $21$.

Question

Which of the following statements is false?

1. Any abelian group of order $27$ is cyclic.

2. Any abelian group of order $14$ is cyclic.

3. Any abelian group of order $21$ is cyclic.

4. Any abelian group of order $30$ is cyclic.

For $2$, it has elements of order $2$ and $7$ and hence an element of order $14$. For $3$ it has elements of order $3$ and $7$ and hence $21$. For $4$, it has elements of order $2,3,$ and $5$ and hence an element of order $30$. So, $2,3,$ and $4$ are all cyclic groups. So I guess $1$ is false. Please help.

Answer 1

The first statement may be false since we can consider the additive abelian group $$\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}.$$ This has 27 elements, but certainly isn't cyclic.