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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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