It is currently Thu Jun 20, 2019 12:21 pm

 All times are UTC [ DST ]

 Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Author Message
 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.

Top

 Post subject: Re: On Solvable Sylow groupsPosted: Wed Sep 07, 2016 12:17 pm
 Team Member

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

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

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 2 posts ]

 All times are UTC [ DST ]

#### Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ Algebra    Linear Algebra    Algebraic Structures    Homological Algebra Analysis    Real Analysis    Complex Analysis    Calculus    Multivariate Calculus    Functional Analysis    Measure and Integration Theory Geometry    Euclidean Geometry    Analytic Geometry    Projective Geometry, Solid Geometry    Differential Geometry Topology    General Topology    Algebraic Topology Category theory Algebraic Geometry Number theory Differential Equations    ODE    PDE Probability & Statistics Combinatorics General Mathematics Foundation Competitions Archives LaTeX    LaTeX & Mathjax    LaTeX code testings Meta