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Questions tagged [fitting-subgroup]

The Fitting subgroup of a group G is the union of all nilpotent normal subgroups of G. It is always a characteristic subgroup. There is also a generalized Fitting subgroup, which is the product of the Fitting subgroup and the layer of the group. To be used with the tag [group-theory].

The Fitting subgroup of a group $G$ is the union of all nilpotent normal subgroups of $G$. See Wikipedia for more information and references.

69 questions
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The "architecture" of a finite group

I think that the aim of finite group theory is the following: Given an arbitrary finite group $G$, study completely the subgroup structure of $G$. There are at least two ways to achieve this purpose: 1) The approach with simple groups. Thanks to…
Dubious
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Solvable group determined by the action on its Fitting subgroup and the isoclass of some of its Sylows

Let Q ≤ P be finite p-groups, H ≤ Aut(Q). Is it really true that there is at most one p-solvable group G such that $Q \unlhd G$, $C_G(Q) \leq Q$, P is a Sylow p-subgroup of G, and the map from G to Aut(Q) surjects onto H? I think this is true (up…
Jack Schmidt
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The Fitting subgroup centralizes minimal normal subgroups in finite groups

Let $G$ be a finite group: If $N$ is a minimal normal subgroup of $G$, then $F(G) \leq C_G(N)$. Here $C_G(N)$ denotes the centralizer of $N$ in $G$, and $F(G)$ denotes the Fitting subgroup of $G$.
M.Mazoo
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Fitting subgroup of wreath product of $\Bbb Z_p$ with an infinite abelian $p$-group.

Let say I have an infinite elementary abelian $p$-group $E$ (i.e. with presentation $E= \langle x_1,x_2,x_3,... \mid x_i^p=1, \ x_i x_j = x_j x_i \rangle$). How do I find the Fitting subgroup of the wreath product $G := \mathbb{Z}_p \wr E$, or…
burek
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Property of Fitting subgroup: Let $G$ be a finite group and $C:= C_G(F(G))$. Then $O_p(C/C\cap F(G))=1$ for every prime $p$.

I'm trying to understand the proof of the following which is stated in Kurzweil and Stellmacher: Let $G$ be a finite group and $C:= C_G(F(G))$. Then $$O_p(C/C\cap F(G))=1$$ for every prime $p$. Here, $F(G)$ is the Fitting subgroup of $G$, i.e.,…
Guest
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Subgroups of smallest possible index in a solvable group

The following question appears in Isaacs' Finite Group Theory: 3B.15) (Berkovich) Let $G$ be solvable, and let $H
user223267
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If $G$ is nonabelian & solvable s.t. the centralizer of each nontrivial element is abelian, then $G$ is Frobenius with kernel its Fitting subgroup

I'm dealing with the following problem in Isaacs Finite Group Theory [6A.5], I would appreciate if you could help: Let $G$ be a nonabelian solvable group in which the centralizer of every nonidentity element is abelian. Show that $G$ is a Frobenius…
Yılmaz
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Subnormal subgroups of the generalized Fitting subgroup

Let $G$ be some finite group. A component of a finite groups is a quasisimple subnormal subgroup. The layer $E(G)$ is the subgroup generated by all components of $G$, let $F(G)$ denote the Fitting subgroup, then $$ F^*(G) = F(G) E(G) $$ is called…
StefanH
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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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Showing all p-local subgroups are char p given that all are contained in char p locals

Question: How does one prove that if every $p$-subgroup $U$ is contained as a subnormal subgroup of a characteristic-$p$, radical, local subgroup containing the normalizer of $U$, then the normalizer of every $p$-subgroup $U$ has characteristic…
Jack Schmidt
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Is the Fitting subgroup compatible with surjective homomorphisms?

If $G$ is a group, its Fitting subgroup $F_G$ is generated by all normal nilpotent subgroups of $G$. By the Fitting Lemma, it is equal to the union of the normal nilpotent subgroups. If $G$ is finite then $F_G$ is also nilpotent but that doesn't…
Martin Brandenburg
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Gorenstein's proof of the classification of solvable CN-groups

I am reading Gorenstein's Finite Groups. Chapter 14 is about CN-groups, a (finite) group where the centralizer of every non-identity element is nilpotent. Theorem 14.1.5 gives the classification of solvable CN-groups, which states that any such…
Ted
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Generalizations of fitting subgroup

The Fitting subgroup of a group $G$ has two generalizations: the generalized Fitting subgroup $F^*(G)$ of Bender and $\tilde F(G)$ of Schmid. The latter is defined by $\tilde F(G)/\Phi(G) = \operatorname{Soc}(G/\Phi(G))$. I saw these in V.I.…
Asieh Hojati
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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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Component of centralizer of involution normalizes component of group

A component of the finite group $G$ is a subgroup $H$ satisfying the following conditions: (1) $H$ is quasisimple (i.e. $H/Z(H)$ is simple and $[H,H]=H$) (2) $H$ is subnormal in $G$ Exercise 6.5.2 in Kurzweil & Stellmacher, The Theory of Finite…
Alex Eustis
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