TITLE: How can I prove that every group of $N = 255$ elements is commutative?
QUESTION [18 upvotes]: There was previous task was same but with $N = 185$. And I prove it by showing that number of Sylow subgroups is 1 for every prime $p\mid N$. But there I have some options $N_5 \in \{1, 51\}$, $N_{17} = 1$, $N_3 \in \{1, 85\}$.
I've tried to get contradiction from $N_5 = 51$ or $N_3=85$, but I didn't manage to do it
I understand that it's impossible to have $N_5 = 51$ and $N_3=85$ at the same time.

REPLY [16 votes]: Suppose $G$ is a finite group of order $|G| = 255 = 3 \cdot 5 \cdot 17$. We will show that $G$ must be cyclic.
As you noticed, there must be only one Sylow subgroup $P$ of order $17$. Thus $P$ must be a normal subgroup. Since $17$ is prime the subgroup $P$ is cyclic, say $P = \langle a \rangle$. It is not difficult to see that every group of order $15$ is cyclic, so $G/P = \langle bP \rangle$ is also cyclic. Now $(bP)^{|b|} = P$, so $|bP| = 15 \mid |b|$. 
If $|b| = 255$ we are done. Therefore we can assume $|b| = 15$. We will show that $ab = ba$ which implies that $ab$ has order $255$ since $15$ and $17$ are coprime. Because $P$ is normal, $bab^{-1} = a^i$ for some integer $i$. Using this fact we get $b^2ab^{-2} = ba^ib^{-1} = (bab^{-1})^i = a^{i^2}$ and by induction $b^kab^{-k} = a^{i^k}$ for all $k \geq 1$. Thus $a = b^{15}ab^{-15} = a^{i^{15}}$, which gives $i^{15} \equiv 1 \mod{17}$. By Fermat's little theorem $i^{16} \equiv 1 \mod{17}$ and since $15$ and $16$ are coprime, $i \equiv 1 \mod{17}$. Therefore $bab^{-1} = a$, proving the claim.
This exercise is a special case of a more general fact. Using the idea of this solution and the fact that in a group of squarefree order the Sylow subgroup corresponding to the largest prime is normal, you can prove that every group of order $n$ is cyclic when $n$ and $\varphi(n)$ are coprime ($\varphi$ is the totient function). This is done by Tibor Szele in the following short article (in German).

T. Szele, Uber die endlichen ordnungszahlen zu denen nur eine Gruppe gehirt, Commentarii Mathematici Helvetici, 20, 265-67, (1947).