# Let $G$ a group and $a\in G$ such that $a$ is an element of finite order, then $\lvert \langle a \rangle\rvert=o(a)$.

I have a theorem to establish:

Let $$G$$ a group and $$a\in G$$ such that $$a$$ is an element of finite order, then $$\lvert \langle a \rangle\rvert=o(a)$$.

We denote $$o(a)$$ as the order of $$a$$.

So my question is...

Is it true that $$G= \langle a \rangle$$ if and only if $$\lvert G \rvert=o(a)$$?

I am getting confused if it follows by definition or it requires a formal proof.

THANKS, for the help :D

• I'm not sure what you mean. It is always true that $| \langle a \rangle | = o(a)$. Do you mean to ask if $G = \langle a \rangle$ if and only if $| \langle a \rangle | = o(G)$? Oct 13, 2020 at 19:38
• No, $|\langle a \rangle|$ is how we define $o(a)$ for $a \in G$, regardless of whether the group $G$ is cyclic. Oct 13, 2020 at 19:38
• In your definition $o(a)$ is defined only if it is finite. If it is finite and $a\in G$ and |$<a>|=|G|$ then $G=<a>$.
– Mark
Oct 13, 2020 at 19:44
• I already do the correction to the text. Oct 13, 2020 at 19:45
• Yes @HallaSurvivor I tried to say $|G|=o(a)$ Oct 13, 2020 at 19:48

You are correct. If for some $$a \in G$$ we have $$|G| = o(a) < \infty$$, then we must have $$G = \langle a \rangle$$.

To see why, we can use containment and finiteness.

Can you show $$\langle a \rangle \subseteq G$$? This will use the fact that $$G$$ is closed under its multiplication, and $$a \in G$$. If you want to be extra formal, you might show each $$a^n \in G$$ by induction on $$n$$.

Next, we use a crucial fact about finite sets. If $$|X| = |Y| < \infty$$ and $$X \subseteq Y$$, then $$X = Y$$. That is, when we're in the finite world, you cannot pull any hilbert's hotel type tricks. So if $$X \subseteq Y$$ and they are the same size, they must actually be the same.

But we showed earlier that $$\langle a \rangle \subseteq G$$, and we are assuming that $$|G| = | \langle a \rangle | < \infty$$. So $$G = \langle a \rangle$$.

I hope this helps! ^_^

• Yes! I alredy understood! Thanks @HallaSurvivor Oct 13, 2020 at 20:22