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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Let $G$ be a finite non-solvable group, each of whose proper subgroups is solvable.

Show that $G/\Phi(G)$ is a non-abelian simple group, where $\Phi(G)$ denotes the Frattini subgroup of $G$ So $G/\Phi(G)$ can't be abelian since if it were then is would be solvable and since $\Phi(G)$ ...
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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 $G$ ...
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Frattini subgroup of $p$-groups and the automorphisms

About $p$-groups, I saw on Wikipedia that: Every automorphism of $G$ induces an automorphism on $G/Φ(G)$, where $Φ(G)$ is the Frattini subgroup of $G$. The quotient $G/Φ(G)$ is an elementary ...
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Frattini sugroup and normal subgroup

For any group $G$, let $\Phi(G)$ denote the Frattini subgroup of $G$. Let $G$ be a finite group, such that $\dfrac{G}{\Phi(G) } \cong K \times \mathbb{Z}_{p}$, where $p$ is prime number. Prove that ...
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How to define a finitely presented group in GAP

In the Group $G=\langle a,b: a^{16}=1 , b^4=a^8 , [a,b]=a^{-2} \rangle$ , the center and the Frattini subgroups are cyclic. I would like to show this fact by using GAP. How I can define this group by ...
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A-invariant Sylow p-subgroup

This is a article which Antonio Beltran. I'm reading lemma 2.2.c). I see that: "Lemma 2.2. Suppose that A is a finite group acting coprimely on a finite group G, and let $C = C_G(A)$. Then, for every ...
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A Supplemented Frattini chief factor of a group

Let $G$ be a group and $H/K$ be a supplemented chief factor of $G$. I have observed one thing: "If $H/K$ is a Frattini chief factor of $G$ (that means, $H/K\leq \Phi(G/K)$) then $H/K$ can be ...
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The Frattini subgroup of $\Bbb{Z}_p \times\Bbb Z _{p^2}.$

Can anyone please help me to find the Frattini subgroup of $\mathbb{Z}_p \times \Bbb Z _{p^2}$? I know that as a set the Frattini subgroup is the set of all non-generators. Is this the only way to ...
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Chief factor of a group and what are its types?

Let $G$ be a group. A section $H/K$ where $H$ and $K$ are normal subgroups of $G$ and $K\leq H$ called a chief factor if $H/K$ is a minimal normal subgroup of $G/K$. What are the possible types of ...
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Generators in a $2$-generated $p$-group

I suppose to have a 2-generated $p$-group $G$. I know that if $\langle a,b\rangle =G$, then $\langle aΦ(G), bΦ(G)\rangle = G/Φ(G)$, where $Φ(G)$ is the Frattini subgroup of $G$. Is it also true that ...
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Maximal subgroups of $p$-groups

The following is exercise 27 from section 6.1 in Dummit and Foote (3rd edition): Let $P$ be a $p$-group and let $\overline{P} = P/frat(P)$ be elementary abelian of order $p^r$. Prove that $P$ has ...
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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 ...
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Is the Frattini subgroup a normal subgroup? [duplicate]

Ive been trying to attack this question from different point of view but i cant make it. Basicly I started thinking that Frattini was not normal, i was trying to get a counterexample but all the ...
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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 $G$ A ...
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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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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 ...
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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 ...