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 Post subject: On Solvable Sylow groupsPosted: Sun Sep 04, 2016 11:15 am

Joined: Mon May 30, 2016 9:19 pm
Posts: 10
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.

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 Post subject: Re: On Solvable Sylow groupsPosted: Wed Sep 07, 2016 12:17 pm
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Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
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

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

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