I am not sure about this. I understand it when it is cyclic. But if not stated so, I cannot reason as to why.($G$ here I assume finite)
$G$ is a group with some subgroup $H$. Then, if $|H|=n$ then there is an element $g \in G$ with order $n$.
Is this true? Then why? I ask this because depending on the sources, when I look at Lagrange's theorem they use it sometimes to say that "there is an element of order $n$ when $|H|=n$ exists in $G$" or indicates so.
I mean, if $H$ is cyclic and has $n$ elements, then it simply means the generator $g$ will have to be multiplied $n$ times to reach around all elements, so $g^n=e$ is understandable. But otherwise...I'm not so sure. Is there a theorem about this?