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RESEARCH PRODUCT

Two Questions of L. A. Shemetkov on Critical Groups

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Throughout the paper we consider only finite groups. Let X be a class of groups. A group G is called s-critical for X , or simply X-critical, if G is not in X but all proper subgroups of G are in X. w Ž .x Ž . Following Doerk and Hawkes 3, VII, 6.1 , we denote Crit X the class s of all X-critical groups. Knowledge of the structure of the groups in Ž . Crit X for a class of groups X can often help one to obtain detailed s information for the structure of the groups belonging to X. Ž w Ž .x. O. J. Schmidt see 5, III, 5.2 studied the N-critical groups, where N is the formation of the nilpotent groups. These groups are also called w x Schmidt groups. In 2 , answering to a question posed by Shemetkov in the w x Kourovka Notebook 6, p. 84 , the authors characterized those subgroupclosed saturated formations f of finite groups such that every f-critical group is either a Schmidt group or a cyclic group of prime order. We shall say that a saturated formation f has the Shemetkov property if every f-critical group is either a Schmidt group or a cyclic group of prime order. Since the structure of the Schmidt groups is well known, the structure of the f-critical groups, where f is a saturated formation with the Shemetkov property, is determined as well. w x Shemetkov 7, Problem 10.22 proposes the following question: ‘‘Let f be a non-empty subgroup-closed formation of finite groups. Assume that f has the Shemetkov property. Is f local?’’ w x Skiba 8 answers this question affirmatively in the soluble universe. He proves that if f is a non-empty subgroup-closed formation of soluble groups with the Shemetkov property, then f is saturated or, equivalently, local. We prove here that this result does not remain true in the general case Ž . Example 2 . In fact, in Theorem 2, we give a criterion for a subgroup-closed