I have a question. Unfortunately, I am not THAT algebra specialist, but I hope someone could help me.
My question is:
Does the following statement hold: "A group of order $n$ is cyclic if and only if it has an element of order $n$."
Well, it's clear that if a group $G$ is cyclic of order $n$, then the order of the generator $g$ has to be $n$, since
$$\vert g\vert=\vert\langle g\rangle\vert =\vert G\vert =n.$$
My problem is that I know what holds if a group is cyclic, but not what has to hold that a group becomes cyclic.
Best Answer
If there is an element $a$ of order $n$, it means that the subgroup $A\leq G$ generated by $a$ is of order $n$. As the order of $G$ is also $n$, you get $A = G$, i.e. $G$ is cyclic and generated for example by $a$.