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Let $p, q$ be prime numbers such that $p<q$ and let $G$ be a group such that $\left| G \right| =pq$.

1. Prove that there exists a unique subgroup $H$ such that $\left| H \right| = q$.
2. Prove that $H$ is a normal subgroup of $G$.
3. Examine if $G$ is a solvable group.

Hello.

i. Firstly, there exist \(\displaystyle{q}\) - Sylow subgroups of \(\displaystyle{G}\).

If \(\displaystyle{n_{q}}\) measures the \(\displaystyle{q}\) - Sylow subgroups of \(\displaystyle{G}\), then,

\(\displaystyle{n_{q}\equiv 1 mod(q)}\) and \(\displaystyle{n_q\mid p}\), so \(\displaystyle{n_q=1}\).

ii. We have that \(\displaystyle{[G:H]=p}\) and \(\displaystyle{p}\) is the smallest prime number

which divides \(\displaystyle{|G|=p\,q}\), so \(\displaystyle{H\triangleleft G}\).

iii. The answer is "YES" . Consider the solvable series

\(\displaystyle{\left\{e\right\}\triangleleft H\triangleleft G}\) and it holds

\(\displaystyle{H/\left\{e\right\}\cong H\cong \mathbb{Z}_{q}\,\,,G/H\cong \mathbb{Z}_{p}}\).