Geodesic
Generalized Frattini Subgroups as Coradicals of Groups
Matematičeskie zametki, Tome 87 (2010) no. 3, pp. 402-411.

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The paper deals with finite solvable groups only. It is established that the class of all regular subgroup $m$-functors coincides with the class of all $X$-abnormal $m$-functors, where $X$ ranges over all subclasses of the class of all primitive groups. The properties of the lattice of all regular subgroup $m$-functors are studied and the atoms and coatoms of this lattice are described. It is proved that the generalized Frattini subgroup of $G$ corresponding to a regular $m$-functor coincides with the $X$-coradical of $G$ for some $R_0$-closed class $X$.
Mots-clés : finite solvable group
Keywords: Frattini subgroup, regular subgroup $m$-functor, Boolean lattice, primitive group, formation of groups, primitivator. 
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S. F. Kamornikov. Generalized Frattini Subgroups as Coradicals of Groups. Matematičeskie zametki, Tome 87 (2010) no. 3, pp. 402-411. http://geodesic.mathdoc.fr/item/MZM_2010_87_3_a7/

[1] R. Carter, T. Hawkes, “The $\mathfrak F$-normalisers of a finite soluble group”, J. Algebra, 5:2 (1967), 175–202 | DOI | MR | Zbl

[2] L. A. Shemetkov, “Stupenchatye formatsii grupp”, Matem. sb., 94:4 (1974), 628–648 | MR | Zbl

[3] S. F. Kamornikov, M. V. Selkin, Podgruppovye funktory i klassy konechnykh grupp, Belorusskaya nauka, Minsk, 2003

[4] L. A. Shemetkov, Formatsii konechnykh grupp, Sovremennaya algebra, Nauka, M., 1978 | MR | Zbl

[5] K. Doerk, T. Hawkes, Finite Soluble Groups, de Gruyter Exp. Math., 4, Walter de Gruyter, Berlin–New York, 1992 | MR | Zbl

[6] R. Baer, “Classes of finite groups and their properties”, Illinois J. Math., 1 (1957), 115–187 | MR | Zbl

[7] L. A. Shemetkov, A. N. Skiba, Formatsii algebraicheskikh sistem, Sovremennaya algebra, Nauka, M., 1989 | MR | Zbl

[8] O. Ore, “Contributions to the theory of groups of finite order”, Duke Math. J., 5:2 (1939), 431–460 | DOI | MR | Zbl