Consider the following statements: $S_1:$ If a group $(G,*)$ is of order $n$, and $a\in G$ is such that $a^m=e$ for some integer $m\le n$, then $m$ must divide $n$. $S_2:$ If a group $(G,)$ is of even order, then there must be an element $a\in G$ such that $a\ne e$ and $aa=e$. Which of the statements is/are correct?

For $S_1$:

By Lagrange theorem, the order of every element of a finite group divides the order of the group.

So, if $m$ is the order of element $a$, then $m$ divides $n$.

Hence, $S_1$ is true.

For $S_2$:

If a finite group has even order, then there exists at least one non-identity element of order $2$.

So, there exists $a\ne e$ such that:

$a*a=e$

Hence, $S_2$ is also true.