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On two problems by L. A. Shemetkov. (English. Russian original) Zbl 0851.20012
Sib. Math. J. 35, No. 4, 713-721 (1994); translation from Sib. Mat. Zh. 35, No. 4, 801-812 (1994).
Let $$\mathfrak F$$ be a nonempty class of groups. A group $$G$$ is called a minimal non-$$\mathfrak F$$-group if $$G$$ does not belong to $$\mathfrak F$$, whereas all proper subgroups of $$G$$ do. When considering problems on enumeration of formations with a prescribed property, L. A. Shemetkov posed the following problem “Let $$\mathfrak F$$ be a nonempty hereditary formation. Assume that every minimal non-$$\mathfrak F$$-group either is a Shmidt group or has prime order. Is $$\mathfrak F$$ local?” L. A. Shemetkov’s problem stated above can be settled in the affirmative in the case when $$\mathfrak F$$ is solvable. However, this is not true in the general case. In the present article we construct an example of a nonlocal hereditary formation $$\mathfrak F$$ that possesses the indicated property for minimal non-$$\mathfrak F$$-groups. At the same time we are able to obtain essential information on the structure of the formation $$\mathfrak F$$ itself. Theorem 1. Let $$\mathfrak F$$ be a nonempty hereditary formation. If every minimal non-$$\mathfrak F$$-group either is a Shmidt group or has prime order then $$\mathfrak F$$ is a composition formation. The proof of Theorem 1 is constructive.

##### MSC:
 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20F17 Formations of groups, Fitting classes
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##### References:
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