Why can't a group with more than p-1 elements be cyclic?

Question

Let p be any element in a prime. Suppose a group G has more than p-1 elements of order p.

Why can't the group be cyclic?

Any hint is appreciated. Thanks in advance.

Answer 1

Yes, this can be solved using basic cyclic groups' lemmas, and the most important for us now is:

Lemma: If $\;G\;$ is a finite cyclic group, then there is one unique subgroup of order any divisor of $\;|G|;$ .

Thus, clearly $\;p\,\mid|G|\;$ by the given data. If $\;G\;$ was cyclic it'd have one unique subgroup of order $\;p\;$ . Finish now the argument...

Answer 2

Sketch: Suppose (for contradiction) that $G$ is cyclic. Let $x$ be order $p$, then $\langle x\rangle$ is a subgroup of order $p$ with exactly $p-1$ elements of order $p$. Since there are more than $p-1$ elements of order $p$, let $y\not\in\langle x\rangle$ be order $p$. Then $\langle y\rangle$ is also a subgroup of order $p$ (and is not equal to $\langle x\rangle$). This contradicts the uniqueness of subgroups of order $p$ in the cyclic group theorems.