Cfrgtkky

I am proving that a group of order $textrm {G}$ being 65 would imply that $textrm {G}$ is cyclic, using Lagrange's Theorem and $N textrm{by} C$ Theorem.

Clearly, one can show that not all non-identity elements in $textrm{G}$ would have order 13. But I have difficulty in proving that not all non-identity elements in $textrm{G}$ would have order 5 either.

Assuming that all non-identity elements in $textrm{G}$ have order 5, the following statement can be concluded:

1) Every non-trivial, proper subgroup of $textrm{G}$ is a cyclic group of order 5.

2) None of those groups will be normal subgroups of $textrm{G}$; otherwise it will imply that $textrm{G}$ will have a subgroup of order 25 which is not possible since $|textrm{G}| = 65$.

3) Index of every non-trivial, proper subgroup of $textrm{G}$ will be 13.

My hunch is that point (3) would possibly show that $textrm{G}$ doesn't have every non-identity element of order 5. But I have difficulty in showing that given any non-trivial, proper subgroup, I can construct more than 13 coset of that subgroup. Or is their any other fact that I'm missing.

General Result:

Let $G$ be a group and $vert Gvert = pq$, where $p$ and $q$ are
distinct primes with $p < q$ and $p nmid q-1$. Then $G$ is cyclic.

Proof. Let $n_p$ be the number of Sylow $p$-subgroups of $G$. Then, by Sylow's Theorem, $n_p mid q$ and $n_p = 1pmod p$. This implies
that either $n_p = 1$ or $n_p = q$. Firstly, suppose that $n_p = q$.
Then $q = 1pmod p$, and so $p mid q-1$ - a contradiction!

Thus $n_p = 1$, and so if $P$ is a Sylow $p$-subgroup of $G$, then $P$
is unique. Thus $P trianglelefteq G$, and by the same token $Q
trianglelefteq G$, where $Q$ is the unique Sylow $q$-subgroup of $G$.
Because $vert Pvert = p$ and $vert Qvert = q$, where $p$ and $q$
are primes, we can write $P = langle x rangle$ and $Q = langle y
rangle$ for some $x,yin G$ with $o(x) = p$ and $o(y)=q$. Also, since
$p$ and $q$ are distinct, $P cap Q = lbrace 1rbrace$ by Lagrange's
Theorem. Now the commutator $[x,y] = x^{-1}y^{-1}xy in Pcap Q = lbrace 1 rbrace$
(because $P$ and $Q$ are normal in $G$). Thus $x^{-1}y^{-1}xy = 1$,
i.e. $xy = yx$. Then it follows easily that $o(xy) = pq$, and so $G =
langle xy rangle$ is cyclic.

In your example, $vert G vert = 65 = 5cdot 13$, so choose $p = 5$ and $q = 13$. Clearly $5 < 13$ and $5 nmid 12$ so the hypothesis of the above result are satisfied. Then $G$ is cyclic.

Hope this helps.

For further information on Sylow's Theorems: https://en.wikipedia.org/wiki/Sylow_theorems