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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. Murashka, A.F. Vasil,ev, On partially conjugate-permutable subgroups of finite groups, but I do not have access to the sources presented in this paper.

I was wondering if you'd mind introducing me to English language resources on these subgroups.

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    $\begingroup$ It would be helpful if you were to define $F^{\ast}(G)$ and $F^{\sim}(G)$...although if you cannot then that is understandable! (Although giving at least $F^{\ast}(G)$ would be ideal...) $\endgroup$ – user1729 Jun 18 '13 at 10:43
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    $\begingroup$ $F^*(G)$ is the generalized Fitting subgroup which is used a lot in technical finite group theory, and in the classification of finite simple groups in particular. I have no idea what $F ^\tilde\,(G)$ is! $\endgroup$ – Derek Holt Jun 18 '13 at 12:36
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    $\begingroup$ @Asieh: you should indicate where you saw these notations. Give the title and author. $\endgroup$ – Jack Schmidt Jun 18 '13 at 13:45
  • $\begingroup$ Answered at mathoverflow.net/questions/135981/… $\endgroup$ – Jack Schmidt Jul 10 '13 at 14:58
  • $\begingroup$ I disagree with this question being closed, and so have voted to re-open (on principle). Notation is difficult, and really this question was closed because noone here knew the answer. Which is a bad reason for closing! $\endgroup$ – user1729 Jul 10 '13 at 18:55
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Overview

Fitting subgroups are designed to be a non-identity ignorable subgroup of a finite group.

The generalized Fitting subgroup was introduced by Bender (1970).

The fundamental property is that $C_G( F^*(G) ) = Z(F^*(G)) = Z(F(G)) \leq F^*(G)$. For solvable groups, the corresponding result is $C_G( F(G) ) = Z(F(G)) \leq F(G)$, and $F(G)$ is minimal amongst all normal subgroups $N$ (of a solvable group) with $C_G(N) \leq N$.

$F^*(G)$ can be defined as the set of elements inducing inner automorphisms on chief factors, and $F(G)$ can be defined as the set of elements centralizing chief factors. In solvable groups, all chief factors are abelian, and so inner=central.

However, in solvable groups there is also a nice feature: the Fitting subgroup is the set of elements centralizing the complemented chief factors. In other words, one can ignore all Frattini factors, especially the Frattini subgroup. This latter version is often written as $F(G/\Phi(G)) = F(G)/\Phi(G)$.

In insoluble groups, this is no longer true. I have been studying the local version of this, as in Lafuente–Martínez-Pérez (2000). The global version is Schmid (1972)'s $\tilde F(G)$ (denoted by $M$). His subgroup is in general larger than $F^*(G)$ and simply forces the matter: $\tilde F(G)/\Phi(G) = F^*( G/\Phi(G) )$. Of course, $\tilde F(G/\Phi(G)) = \tilde F(G) / \Phi(G)$.

References

The view that the generalized Fitting subgroup is defined in terms of chief factors and quasinilpotent subgroups is in Huppert-Blackburn X.13. It also covers the traditional component+Fitting definition. Aschbacher 11.31 and Kurzweil–Stellmacher 6.5 only covers the traditional approach but are highly recommended. If you like the quasinilpotent approach, then see also example IX.2.5.c page 579-581 in Doerk–Hawkes.

Schmid's subgroup is also important, but I have found fewer sources on it. Aschbacher's textbook does not seem to discuss it. Ezquerro–Ballester-Bolinches uses it (denoted by $F'(G)$) in a critical way in Definition 1.4.9.

Both $F^*$ and a slight variation $\tilde F$ are discussed in I.1A.4 of Gorenstein-Lyons-Solomon. See 4.10 for the $\tilde F$ variation, and its very similar property. This entire section explains the point of the various Fitting subgroups.

Articles

  • Bender, Helmut. “On groups with abelian Sylow 2-subgroups.” Math. Z. 117 (1970) 164–176. MR288180 DOI:10.1007/BF01109839
  • Schmid, Peter. “Über die Automorphismengruppen endlicher Gruppen.” Arch. Math. (Basel) 23 (1972), 236–242. MR308266 DOI:10.1007/BF01304876
  • Lafuente, Julio P.; Martínez-Pérez, Conchita. “$p$-constrainedness and Frattini chief factors.” Arch. Math. (Basel) 75 (2000), no. 4, 241–246. MR1786169 DOI:10.1007/s000130050499

Books

  • Aschbacher, Finite Group Theory (1988)
  • Doerk–Hawkes, Finite Soluble Groups (1992)
  • Ezquerro–Ballester-Bolinches, Class of Finite Groups (2006)
  • Gorenstein-Lyons-Solomon, The Classification of the Finite Simple Groups (1994)
  • Huppert-Blackburn, Finite Groups, volume 3 (1982)
  • Kurzweil-Stellmacher, The Theory of Finite Groups (2004)
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