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The [[Characteristic subgroup|characteristic subgroup]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f0405101.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f0405102.png" /> generated by all the nilpotent normal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f0405103.png" />; it is also called the Fitting radical. It was first studied by H. Fitting [[#References|[1]]]. For finite groups, the Fitting subgroup is nilpotent and is the unique maximal nilpotent normal subgroup of the group. For a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f0405104.png" /> the following relations hold:
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The [[Characteristic subgroup|characteristic subgroup]] $F(G)=F$ of a group $G$ generated by all the nilpotent normal subgroups of $G$; it is also called the Fitting radical. It was first studied by H. Fitting [[#References|[1]]]. For finite groups, the Fitting subgroup is nilpotent and is the unique maximal nilpotent normal subgroup of the group. For a finite group $G$ the following relations hold:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f0405105.png" /></td> </tr></table>
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$$ [F,F]\subseteq\Phi\subseteq F \quad and \quad F/\Phi=F(G/\Phi) , $$
 
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where $\Phi$ denotes the [[Frattini-subgroup(2)|Frattini subgroup]] of $G$, and $[F,F]$ is the [[Commutator subgroup|commutator subgroup]] of $F$.
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f0405106.png" /> denotes the [[Frattini-subgroup(2)|Frattini subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f0405107.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f0405108.png" /> is the [[Commutator subgroup|commutator subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f0405109.png" />.
 
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
A finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051010.png" /> is called quasi-nilpotent if and only if for every chief factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051012.png" /> every automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051013.png" /> induced by an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051014.png" /> is inner. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051015.png" /> be the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051016.png" />. Inductively define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051018.png" /> by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051019.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051020.png" /> are all normal subgroups. The series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051021.png" /> is the ascending central series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051022.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051023.png" />, the so-called hypercentre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051024.png" />. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051025.png" /> is semi-simple if and only if it is the direct product of non-Abelian simple groups. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051026.png" /> is quasi-nilpotent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051027.png" /> is semi-simple. The generalized Fitting subgroup of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051028.png" /> is the set of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051030.png" /> which induce an inner automorphism on every chief factor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051031.png" />. It is a characteristic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051032.png" /> and contains every subnormal quasi-nilpotent subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051033.png" />. This property can therefore also be used to define it.
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A finite group $G$ is called quasi-nilpotent if and only if for every chief factor $H$ of $G$ every automorphism of $H$ induced by an element of $G$ is inner. Let $Z(G)=Z_0(G)=\{g\in G:gh=hg\text{ for all }h\text{ such that }h\in G\}$ be the centre of $G$. Inductively define $Z_i(G)\supseteq Z_{i-1}(G)$, $i=1,2,\ldots$ by the condition $Z_i(G)/Z_{i-1}(G)=Z(G/Z_{i-1}(G))$. The $Z_i(G)$ are all normal subgroups. The series $Z_0(G)\subseteq Z_1(G)\subseteq\ldots$ is the ascending central series of $G$. Let $Z_\infty(G)=\bigcup_i Z_i(G)$, the so-called hypercentre of $G$. A group $G$ is semi-simple if and only if it is the direct product of non-Abelian simple groups. A group $G$ is quasi-nilpotent if and only if $G/Z_\infty(G)$ is semi-simple. The generalized Fitting subgroup of a finite group $G$ is the set of all elements $x$ of $G$ which induce an inner automorphism on every chief factor of $G$. It is a characteristic subgroup of $G$ and contains every subnormal quasi-nilpotent subgroup of $G$. This property can therefore also be used to define it.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051034.png" /> be the lower central series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051035.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051036.png" />, the commutator subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051037.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051038.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051039.png" /> be the generalized Fitting subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051040.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051041.png" /> is a perfect quasi-nilpotent characteristic subgroup. It is sometimes called the layer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040510/f04051042.png" />.
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Let $G=\gamma_1(G)\supseteq\gamma_2(G)\supseteq\ldots$ be the lower central series of $G$, i.e. $\gamma_n(G)=[G,\gamma_{n-1}(G)]$, the commutator subgroup of $G$ with $\gamma_{n-1}(G)$. Let $F^\ast(G)$ be the generalized Fitting subgroup of $G$; then $E(G)=\bigcap_{n\geq 1}\gamma_n(F^\ast(G))$ is a perfect quasi-nilpotent characteristic subgroup. It is sometimes called the layer of $G$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Huppert,  "Finite groups" , '''2–3''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.J.S. Robinson,  "A course in the theory of groups" , Springer  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,  "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Huppert,  "Finite groups" , '''2–3''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.J.S. Robinson,  "A course in the theory of groups" , Springer  (1982)</TD></TR></table>

Revision as of 09:28, 10 February 2012

The characteristic subgroup $F(G)=F$ of a group $G$ generated by all the nilpotent normal subgroups of $G$; it is also called the Fitting radical. It was first studied by H. Fitting [1]. For finite groups, the Fitting subgroup is nilpotent and is the unique maximal nilpotent normal subgroup of the group. For a finite group $G$ the following relations hold:

$$ [F,F]\subseteq\Phi\subseteq F \quad and \quad F/\Phi=F(G/\Phi) , $$ where $\Phi$ denotes the Frattini subgroup of $G$, and $[F,F]$ is the commutator subgroup of $F$.

References

[1] H. Fitting, "Beiträge zur Theorie der Gruppen endlicher Ordnung" Jahresber. Deutsch. Math.-Verein , 48 (1938) pp. 77–141
[2] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[3] D. Gorenstein, "Finite groups" , Harper & Row (1968)


Comments

A finite group $G$ is called quasi-nilpotent if and only if for every chief factor $H$ of $G$ every automorphism of $H$ induced by an element of $G$ is inner. Let $Z(G)=Z_0(G)=\{g\in G:gh=hg\text{ for all }h\text{ such that }h\in G\}$ be the centre of $G$. Inductively define $Z_i(G)\supseteq Z_{i-1}(G)$, $i=1,2,\ldots$ by the condition $Z_i(G)/Z_{i-1}(G)=Z(G/Z_{i-1}(G))$. The $Z_i(G)$ are all normal subgroups. The series $Z_0(G)\subseteq Z_1(G)\subseteq\ldots$ is the ascending central series of $G$. Let $Z_\infty(G)=\bigcup_i Z_i(G)$, the so-called hypercentre of $G$. A group $G$ is semi-simple if and only if it is the direct product of non-Abelian simple groups. A group $G$ is quasi-nilpotent if and only if $G/Z_\infty(G)$ is semi-simple. The generalized Fitting subgroup of a finite group $G$ is the set of all elements $x$ of $G$ which induce an inner automorphism on every chief factor of $G$. It is a characteristic subgroup of $G$ and contains every subnormal quasi-nilpotent subgroup of $G$. This property can therefore also be used to define it.

Let $G=\gamma_1(G)\supseteq\gamma_2(G)\supseteq\ldots$ be the lower central series of $G$, i.e. $\gamma_n(G)=[G,\gamma_{n-1}(G)]$, the commutator subgroup of $G$ with $\gamma_{n-1}(G)$. Let $F^\ast(G)$ be the generalized Fitting subgroup of $G$; then $E(G)=\bigcap_{n\geq 1}\gamma_n(F^\ast(G))$ is a perfect quasi-nilpotent characteristic subgroup. It is sometimes called the layer of $G$.

References

[a1] B. Huppert, "Endliche Gruppen" , 1 , Springer (1967)
[a2] B. Huppert, "Finite groups" , 2–3 , Springer (1982)
[a3] D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)
How to Cite This Entry:
Fitting subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fitting_subgroup&oldid=20946
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article