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September 12th, 2017, 11:51 AM   #11
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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.
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September 12th, 2017, 12:49 PM   #12
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What does $\displaystyle \left\{e\phi(G)\right\}$ mean?
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September 12th, 2017, 01:23 PM   #13
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$e$ is the trivial element of $G$
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September 12th, 2017, 05:03 PM   #14
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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.
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September 12th, 2017, 11:16 PM   #15
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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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