Let $p$ be prime. If a group has more than $p-1$ elements of order $p$, why can't the group be cyclic?
Can I assume that the group must be finite group since there is at least one element with finite order?
And I am wondering now if all cyclic groups are finite, or all finite groups are cyclic?
An infinite cyclic group has no elements of finite order, except for $e$, which has order $1$. Thus, your group cannot be infinite cyclic.
A finite cyclic group of order $n$ has exactly one subgroup of order $d$ for each $d$ dividing $n$.
A subgroup of order $p$ in any group has exactly $p-1$ elements of order $p$.
Therefore, if a group has more than $p-1$ elements of order $p$, then it has more than one subgroup of order $p$ and so cannot be cyclic.