Relationship between submonoids and subgroups

Question

I'm taking my first abstract Algebra course. During one of the last lessons, my teacher told us that

If $G$ is a group and $M$ is a finite submonoid of $G$, then $M$ is a subgroup of $G$.

For the properties of submonoids, $M$ is clearly closed under multiplication and $1_G \in M$. It is not clear to me why $x \in M \implies x^{-1} \in M$.

Could somebody give me a hint to come up with a reason for this? Thank you.

Answer 1

Hint: Let $x\in M$ and consider all powers of $x$. Show that $x^n=1$ for some $n$. Then use this to show that $x^{-1}\in M$.

Answer 2

$M$ is finite submonoid.

Thus $\forall x \in M , \ \exists n \in \mathbb N , x^n=1_G$

It's obvious that $x^{-1}=x^{n-1} \in M$