TITLE: Let $p$ be a prime. If a group has more than $p − 1$ elements of order $p$, why can’t the group be cyclic?
QUESTION [8 upvotes]: I need help to prove the following result 
Let $p$ be a prime. If a group has more than $p − 1$ elements
        of order $p$, why can’t the group be cyclic? 
Thanks for the help.

REPLY [3 votes]: If $G$ is infinite cyclic group ,it can't have the element of prime order . 
So suppose $G$ is  finite cyclic group 
which contains some elements of order $p$, $p$ must divide $|G|$ by Lagrange's Theorem. Then by the Fundamental Theorem of Finite Cyclic Groups, for the divisor $p$ of $|G|$, there is exactly one subgroup of order $p$. In that subgroup of order $p$, there are $\phi(p)=p-1$ elements of order $p$. Hence, there are only $p-1$ elements of order $p$ in $G$.
If there are more than $p-1$ elements of order $p$, then there must be one more subgroups of order $p$, which contradicts with the Fundamental Theorem of Finite Cyclic Groups. Hence $G$ is not cyclic.