Geodesic
Soluble formations with the Shemetkov property
Problemy fiziki, matematiki i tehniki, no. 1 (2015), pp. 82-87.

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All saturated soluble formations whose all $s$-critical groups are soluble were described. With every local formation $\mathfrak{F}=LF(f)$, such that $f(p)=\mathfrak{S}_{\pi(f(p))}$ for all $p\in\pi(\mathfrak{F})$ and $f(p)=\varnothing$ otherwise, was associated directed graph $\Gamma(\mathfrak{F},f)$ without loops whose vertices are prime numbers from $\pi(\mathfrak{F})$ and $(p_i,p_j)$ is an edge of $\Gamma(\mathfrak{F},f)$ if and only if $p_j\in\pi(f(p_i))$. With the help of such kind’s graphs all hereditary soluble formations with the Shemetkov property were described.
Keywords: minimal simple group, $s$-critical group, hereditary local formation, formation with the Shemetkov property, graph associated with formation. 
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V. I. Murashka. Soluble formations with the Shemetkov property. Problemy fiziki, matematiki i tehniki, no. 1 (2015), pp. 82-87. http://geodesic.mathdoc.fr/item/PFMT_2015_1_a15/

[1] O. Yu. Shmidt, “Groups all whose subgroups are special”, Mat. Sbornik, 31 (1924), 366–372 (In Russian)

[2] N. Ito, “Note on (LM)-groups of finite order”, Kodai Math. Seminar Report, 3:1–2 (1951), 1–6 | DOI

[3] The Kourovka Notebook: Unsolved Problems in Group Theory, Novosibirsk, 1992

[4] V. N. Semenchuck, A. F. Vasil'ev, “Characterization of local formations $\mathfrak{F}$ by properties of minimal non-$\mathfrak{F}$-groups”, Studies of normal and subgroup structure of finite groups, Nauka i technika, 1984, 175–181

[5] A. N. Skiba, “On a class of formations of finite groups”, Dokl. BSSR, 34:11 (1994), 982–985 (In Russian)

[6] A. Ballester-Bolinshes, M. D. Perez-Ramos, “On $\mathfrak{F}$-critical groups”, J. Algebra, 174 (1995), 948–958 | DOI

[7] S. F. Kamornikov, M. V. Selkin, Subgroup's functors and classes of finite groups, Belaruskaya nauka, Minsk, 2003, 254 pp. (In Russian)

[8] S. F. Kamornikov, “On two problems by L. A. Shemetkov”, Siberian Mathematical Journal, 35:4 (1994), 713–721 | DOI

[9] T. Hawkes, “On the class of Sylow tower groups”, Math. Z., 105 (1968), 393–398 | DOI

[10] K. Doerk, T. Hawkes, Finite soluble groups, Walter de Gruyter, Berlin–New-York, 1992, 891 pp.

[11] B. Huppert, Endlicher gruppen, v. I, Springer, Berlin, 1967, 793 pp.

[12] M. Suzuki, “On a class of doubly transitive groups”, Ann. Math., 75 (1962), 105–145 | DOI

[13] A. N. Skiba, Algebra of formations, Belaruskaya nauka, Minsk, 1997, 240 pp. (In Russian)