My Math Forum action of Frobenius group

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 September 12th, 2017, 11:51 AM #11 Member   Joined: Jan 2015 From: usa Posts: 75 Thanks: 0 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: 33 Thanks: 4 What does $\displaystyle \left\{e\phi(G)\right\}$ mean?
 September 12th, 2017, 01:23 PM #13 Member   Joined: Jan 2015 From: usa Posts: 75 Thanks: 0 $e$ is the trivial element of $G$
 September 12th, 2017, 05:03 PM #14 Member   Joined: Jan 2016 From: Athens, OH Posts: 58 Thanks: 34 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 Member   Joined: Jan 2015 From: usa Posts: 75 Thanks: 0 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.

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