Question

Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that (Q, +) is not cyclic.

Answer 1

Let, $G$ be a finite group of order $pq$ , where $p<q$ are primes.

Recall that the number $t_q$ of sylow $q$ -subgroup of $G$

satisfies $\frac{t_q}{p}$ and $t_q=1$

$if\space t_q\not=1$

Then, $t_q\geqslant q+1>p$ which contradicts the $\frac{t_q}{p}.$

Thus $t_q=1$

Then G has normal sylow q-subgroup Q

we have $\frac{t_q}{p}$

and $t_q=1$

The first condition implies that $t_p=1$ or $t_p=q$

The latter case implies that $q=1$ which is excluded by our assumption that $p=q-1$

Thus, $t_p=1\space and \space G$ has a normal sylow p-subgroup p.