Is the group $\mathbb Z/3 \times \mathbb Z/5$ cyclic?

Question

In understanding if there exist an abelian group of order $15$ that is not cyclic, I came up with this example: the direct product $\mathbb{Z}/3 \times \mathbb{Z}/5$. I know that this group is abelian, but I'm unsure if its cyclic or not.

If I can find one element in this set where I multiply it by a number that is not $15$, and I get $(\overline{0},\overline{0})$ then it can't be cyclic and I'm done, I think this is the case at least, is my logic correct, or would I need some number less than 15, multiply every element in this set by a number strictly less than 15, and get $(\overline{0},\overline{0})$. If this is the case then no such example exists.

There is only one group of order $15$ up to isomorphism. It is cyclic. — Shaun, Oct 10 '21 at 18:49
More generally, for any natural number $n$, if $n$ and $\phi(n)$ are relatively prime, there is only one group of order $n$ up to isomorphism and it is cyclic. — Rushy, Oct 10 '21 at 18:51
$\mathbb{Z}/3 \times \mathbb{Z}/5$ is cyclic by the Chinese Remainder Theorem. — Geoffrey Trang, Oct 10 '21 at 18:56
The linked question and answer involve Sylow’s theorems and other more sophisticated techniques that IMO make it a different kind of question than the one the OP is asking. — Sammy Black, Oct 10 '21 at 19:11
You can try prove that $\mathbb Z / m\mathbb Z \times \mathbb Z / n \mathbb Z$ is cyclic iff $\text{gcd} (m,n) =1$. Try Bezout's lemma. — ashK, Oct 10 '21 at 19:49
Your logic should be:" If I can find an element such that multiplied by a number from $1$ to $14$ I do not get $(0,0)$ then the group is cyclic". There are many such elements, one example is $(1,1)$. — orangeskid, Oct 10 '21 at 19:56
Why did you delete this question? If you are just trying to hide your tracks, then kindly take your efforts elsewhere. — Arturo Magidin, Oct 24 '21 at 04:15

score 2 · Accepted Answer · answered Oct 10 '21 at 19:50

Hint: What is the order of the element $(1,1) \in \mathbb{Z}_3 \times \mathbb{Z}_5?$ (there is a general formula for the order of an element in a direct product, but you can check it what the order is here by hand if you don't know this formula). What can you conclude about the group?

It can be shown that any group of order $15$ is isomorphic to $\mathbb{Z}_{15}$. This is usually proven by Sylow theory.

Is the group $\mathbb Z/3 \times \mathbb Z/5$ cyclic?

1 Answers1