Groups of order $p(p+1)$

Question

If I have a group of order $p(p+1)$ with $p+1$ Sylow $p$-subgroups how can I prove that all $p$ non-trivial elements not of order $p$ have prime order?

Answer 1

Since $G$ has $p+1$ Sylow $p$-subgroups, we have $P = N_G(P)$ for $P \in {\rm Syl}_p(G)$. So if $h \not\in P$, then $h \not\in N_G(P)$, so no nontrivial element of $P$ can centralize $h$. Hence $h$ has $p$ distinct conjugates under the action of $P$. So the $p$ elements that do not have order $p$ must all be conjugate, and so they all have the same order and, since at least one of them has prime order, they all have prime order.

Note also that if the prime order in question is $q$, then $p+1$ is a power of $q$, so the only possibilities are $p=2$, and $p$ a Mersenne prime with $q=2$. For each such $p$ there is a unique isomorphism class of groups of this form.