
Abstract Algebra Abstract Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
September 12th, 2017, 11:51 AM  #11 
Senior Member Joined: Jan 2015 From: usa Posts: 103 Thanks: 1 
I have another question that I can't answer; please help me if you can: let $G$ a group. I want to show that $\phi(G/\phi(G))=\left\{e\phi(G)\right\}$ where $\phi(G)$ is the Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$). Last edited by skipjack; September 13th, 2017 at 02:52 AM. 
September 12th, 2017, 12:49 PM  #12 
Member Joined: May 2017 From: Russia Posts: 34 Thanks: 5 
What does $\displaystyle \left\{e\phi(G)\right\}$ mean?

September 12th, 2017, 01:23 PM  #13 
Senior Member Joined: Jan 2015 From: usa Posts: 103 Thanks: 1 
$e$ is the trivial element of $G$

September 12th, 2017, 05:03 PM  #14 
Member Joined: Jan 2016 From: Athens, OH Posts: 91 Thanks: 47 
First, you should start a new thread when you have a new question. However: For any group G and normal subgroup N of G, the subgroups of G/N are precisely of the form H/N where H is a subgroup of G that contains N. From here, it should be obvious that if $N\subseteq \phi(G)$, then $\phi(G/N)=\phi(G)/N$. In particular, when N is the Frattini subgroup, the Fratinni subgroup of G/N is trivial. 
September 12th, 2017, 11:16 PM  #15 
Senior Member Joined: Jan 2015 From: usa Posts: 103 Thanks: 1 
And how does that answer the question? You showed that $\phi(G/N)=\phi(G)/N$ how can we conclude that $\phi(G/\phi(G))=\left\{e\phi(G)\right\}$? Thanks. Last edited by skipjack; September 13th, 2017 at 02:50 AM. 

Tags 
action, frobenius, group 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Showing that an action of a group on a set is proper  princetongirl818  Abstract Algebra  0  November 24th, 2016 10:37 PM 
Frobenius norm  Maissa  Linear Algebra  2  May 29th, 2011 03:25 AM 
Frobenius method  zokomoko  Applied Math  0  June 14th, 2010 08:38 AM 
group action on set  tinynerdi  Abstract Algebra  4  March 11th, 2010 06:54 AM 
please help quickly about sylow thm and group action quest  bogazichili  Algebra  1  December 13th, 2008 07:13 AM 