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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].

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 .

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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 the maximal subgroups of $G$, and I know that it…
Alex Petzke
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Index of subgroups in a finite solvable group, with trivial Frattini subgroup (Exercise 3B.12 from Finite Group Theory, by M. Isaacs)

Let G be a finite solvable group, and assume that $\Phi(G) = 1$ where $\Phi(G)$ denotes the Frattini subgroup of G. Let M be a maximal subgroup of G, and suppose that $H \subseteq M$. Show that $G$ has a subgroup with index equal to $|M:H|$. This is…
DGB
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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 can be easily proved for $p$-groups by seeing that…
Chain Markov
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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 stick with this example first. I know the…
Nogard
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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 one used in the answer to “If $|G|=p^3q^2$ then…
Chain Markov
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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 solve the problem for the following cases: $P \cong…
Chain Markov
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What if an automorphism fixes every maximal subgroup pointwise. Is it then the identity?

This question came up in the discussion over here My first thought was that then it fixes the Frattini subgroup. Any help? For reference we found that the answer is no when each maximal subgroup is merely mapped back to itself.
user1007655
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Nilpotency class of Frattini subgroup and group order

Suppose $\psi(n)$ denotes the minimal natural number $k$, such that there exists a finite group $G$, such that $k = \max \{m \in \mathbb{N}| \exists \text{ prime } p, p^m | |G| \}$, and $\Phi(G)$ has nilpotency class exactly $n$. Here $\Phi$ stands…
Chain Markov
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Fitting subgroup of a finite solvable group with trivial center and trivial Frattini subgroup.

$\textbf{The question is as follows:}$ Let $G$ be a finite solvable group with trivial center and trivial Frattini subgroup. Show that its Fitting subgroup $F(G)$ is contained in $G'$. $\textbf{This question has a hint:}$ [Hint: show that…
Nikita
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Is there an algorithm to check whether given subgroup contained inside the Frattini subgroup?

I am new to algorithmic group theory. I have the following question: Let $G$ be a group. The Frattini subgroup of $G$ is the intersection of all maximal subgroup of $G$, denoted by $\Phi(G)$. It is also equal to the set of all non-generating…
Pranjal
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Frattini subgroup of a simple non abelian group

How do I calculate the Frattini subgroup of a simple non-abelian group $G$? My efforts: Well, it's a well know fact that $\Phi(G) \trianglelefteq G$ because $\Phi(G)$ is a characteristic subgroup. Thus $\Phi(G) = 1$ or $\Phi(G) = G$ due to the…
nom
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A question related to Frattini subgroup

I am doing a question which need the following statement as a lemma: Statement: If $G$ is a group with a finitely generated Frattini subgroup $\Phi(G)$, then the only subgroup $H$ of $G$ such that $H\Phi(G) = G$ is $H = G$. The proof of this…
Andrew
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Fitting subgroup is abelian and has complement if the mother group is Frattini free

I am currently reading on some theorems about relations between finite group and its largest cardinality of independent generating sequence. One assumed-well-known result is that if given a finite Frattini free group $G$ (Frattini free means the…
S_j
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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 .
user148528
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Finite groups with a certain Frattini subgroup

Let $G$ be a finite group different from a cyclic $p-$group and $\Phi(G)=M_i\cap M_j$, where $M_i$ and $M_j$ are two arbitrary distinct maximal subgroups of $G$ and $i, j \geq 1$. Is it possible to characterize such $G$? ($\Phi(G)$ denotes the…
sebastian
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