# 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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### Finite solvable Frattini-free group having a unique minimal normal subgroup N implies that N is the Fitting subgroup

This is exercise 6.1.6 of Kurzweil and Stellmacher. A restatement is: Let $G$ be a finite solvable group with $\Phi(G)=1$, and assume that $G$ has a unique minimal normal subgroup $N$. Then $N=F(G)$....
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### The number of Hall $\pi$-subgroups of a $\pi$-separable group - Alexandre Turull article

This is an article which Alexandre Turull wrote. Lemma 2.1. states Lemma 2.1. Suppose $H$ is a finite group, acting on the finite group $F$, and assume that $|H|$ and $|F|$ are relatively prime. ...
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### Frattini subgroup of a quotient

Let $G$ be a finite group. The Frattini subgroup $\Phi(G)$ is the intersection all proper maximal subgroups. If $K \lhd G$ is a normal subgroup, then it is easy to see that $\Phi(G) K/K \leq \Phi(G/K)$...
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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 ...
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### Given a finite $p$-group $G$ and $\Phi(G)$ the Frattini subgroup of $G$. Prove $G/\Phi(G)$ is $p$-torsion

As the title says, I need to prove that every element in $G/\Phi(G)$ has order $p$. I know by the First Sylow theorem that $G$ contains subgroups of orders $1,p,p^2,\dots,p^{k-1}$ with each normal in ...
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### The Frattini subgroup of the standard Wreath product of two quasicyclic groups is the group itself.

This is part of Exercise 5.2.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. Here are some previous questions ...
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### What if an automorphism fixes every maximal subgroup pointwise. Is it then the identity? [closed]

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 ...
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### Frattini subgroup of $p$-group

Suppose $G$ is a finite, non-trivial $p$-group and $\Phi(G)$ is the Frattini subgroup, defined as the intersection of all maximal subgroups of $G$. Since $G$ is finite, there are finitely many maximal ...
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### The Automorphisms of a group with a certain presentation

$\textbf{Exercise.}$ If $G=\langle x,y ~|~ x^2=1=y^{2^n},y^x=y^{1+2^{n-1}} \rangle$, prove that $\mathrm{Aut}(G)$ is a $2$-group. I tried using this result, calculated the Frattini subgroup of $G$ ...
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### Frattini subgroup of p-groups characteristics

"Assume $P$ is a $p$-group and $N$ is normal in $P$ with this property that $P/N$ is an abelian elementary group. Prove that $\Phi(P)$ is in $N$. (Note: $\Phi(P)$ is intersection of maximal ...
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### properties of the Frattini subgroup of a finite group

I am reading the following proposition the question is: where do we use the hypothesis that $G$ is a finite group in the first point?
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### Showing $\Phi(G)=G'G^p$.

Prove $$\Phi(G)=G'G^p,$$ where $\Phi(G)$ is the Frattini subgroup of $G$, the intersection of all maximum subgroups of $G$, $G'=[G:G]$ is the commutator subgroup of $G$, and $G^p$ is the group ...
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### If $\theta :G\rightarrow H$ is a surjective homomorphism, then $\theta(\Phi(G))\leq\Phi(H)$.

This is a claim when I try to solve another problem related to the Fratinni group of a p-group, and I saw an answer Frattini subgroup of a finite elementary abelian $p$-group is trivial. I am stuck at ...
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