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Questions tagged [frattini-subgroup]

Frattini subgroup of a group $G$ is the intersection of all maximal proper subgroups of $G$. It can be also equivalently defined as the set of all nongenerators of $G$ ($x \in G$ is a nongenerator of $G$ iff $\forall S \subset G ((\langle S \cup \{x\} \rangle = G) \rightarrow (\langle S \rangle = G))$). Frattini subgroup is always a characteristic subgroup. To be used with the tag [group-theory].

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Characterisation of the Frattini Group

Let $\Phi(G)$ be the set of non-generators of $G$, that is the groups of elements $g\in G$ such that $\langle X,g\rangle=G \implies \langle X\rangle=G$ We call such set the Frattini Subgroup of $G$ ...
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Is this proof that $\widehat{G_p}$ is pro-$p$ free correct?

Let $G$ be an abstract group with the following presentation: $$G \simeq \langle x,y \mid x^2y^2 = 1 \rangle $$ Let $p \neq 2$ be an odd prime. I want to show that $\widehat{G_p} \simeq \mathbb{Z}_p$...
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A question about Frattini subgroup of specific form v2.0

Suppose $p$ is a prime number and $G$ is a finite group, such that $\Phi(G) = D_4 = \langle a \rangle_4 \rtimes \langle b \rangle_2$, where $\Phi$ denotes the Frattini subgroup. Is it always true, ...
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Does non-abelian $\Phi(G)$ imply, that $p^5 | |G|$ for some prime $p$?

Suppose $G$ is a finite group, such that $\Phi(G)$ is non-abelian. Does there always exist such prime $p$, that $p^5 | |G|$? Here $\Phi$ stands for Frattini subgroup. Using the same method, as the ...
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A question about Frattini subgroup of specific form

Suppose $p$ is a prime number and $G$ is a finite group, such that $\Phi(G) = C_p \times C_p$, where $\Phi$ denotes the Frattini subgroup. Is it always true, that $p^4$ divides $|G|$? This statement ...
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Is the intersection of Frattini subgroup and a Sylow subgroup contained in the Frattini subgroup of the Sylow subgroup?

Suppose $G$ is a finite group, $P$ is a Sylow p-subgroup of $G$. Is it always true, that $\Phi(G) \cap P$ is a subgroup of $\Phi(P)$? Here $\Phi(G)$ is the Frattini subgroup of $G$. I managed to ...
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Frattini subgroup of a subgroup

I am doing an exercise for Frattini subgroups: Let $G$ be a finite group and $N$ a normal subgroup of $G$. Show that there exists a subgroup $K$ of $G$ with $G=KN$ and $K \cap N \leq \Phi(K)$. I ...
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If $|G|=p^3q^2$ then $\Phi(G)$ is cyclic for primes $p\neq q$.

I have conjectured this result for the Frattini subgroup by doing some calculations in GAP. I think this is even true if $|G|=p_1^{i_1}\cdots p_n^{i_n}$ for $i_j\leq 3$ holds, but I would like to ...
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About Frattini subgroup

Let $ G $ is a finite group. Suppose $ H \unlhd G $ such that $ G/H $ is supersoluble. Suppose $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup of $ G $. Suppose ...
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A problem about Frattini subgroup of a subgroup

Let $H$ be a subgroup of $G$. Does that imply, that $\Phi(H)\le \Phi(G)$? If not, then what properties $G$ must have for it to be true. $\Phi$ stands for Fattini subgroup .
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Intuition behind the Frattini subgroup

I am trying to get a better feel for what the Frattini subgroup really is, intuitively. Let $G$ be a group and denote its Frattini subgroup by $\Phi(G)$. I know that $\Phi(G)$ is the intersection of ...