Let $G$ be a group, $H$ a subgroup of $G$, and $h \in H$. What is the relation of the order of $h$ in $H$ and the order of $h$ in $G$?
I guess the answer is "the same". Let us denote by $h$ the order of $h$. Then
$(1)$ $|h|=m$ in $H$ $\iff$ $|h|=m$ in $G$.
$(2)$ $|h|=+\infty$ in $H$ $\iff$ $|h|=+\infty$ in $G$.
Because if we denote by $\langle h \rangle$ the cyclic subgroup of $G$ generated by $h$, then $\langle h \rangle \le H \le G$. Moreover, we know $|h|=|\langle h \rangle|$.
What do you think?
Yes this is exactly the same: a subgroup is namely characterized by the fact that it inherits the group structure from its parent group. With quotient groups this is different: an element could have infinite order in the parent group, while it has finite order in the quotient.