Two-step nilpotent FC-groups
The known examples of periodic FC-groups which are not sections of direct products of finite groups (QSD\({\mathfrak F})\) all involve extra-special p- groups. Question 3H of the reviewer's book `FC-groups' [Pitman Res. Notes Math. 96 (1984; Zbl 0547.20031)] therefore asked whether every periodic FC-group is a section of a direct product of groups with finite derived subgroup (QSD\({\mathfrak FA})\). One of the author's results answers this question in the negative. Theorem 2: If the QSD\({\mathfrak FA}\)-group G is nilpotent of class two then G/Z(G) is embeddable in a direct product of finite groups (SD\({\mathfrak F})\). There is an example due to Gorchakov of a periodic FC-group which is nilpotent of class two but for which G/Z(G)\(\not\in SD{\mathfrak F}\). [The author has since removed the nilpotent condition from Theorem 2.] Theorem 4: If the extraspecial p-group G is in the class QSD\({\mathfrak F}\), then G can be embedded in the direct product with amalgamated centre of nonabelian groups of order \(p^ 3\). This answers Question 3G and shows that, in the notation used there, the class of periodic \({\mathfrak Z}\)-groups is strictly larger than the class of QSD\({\mathfrak F}\)-groups.
sections of direct products of finite groups
scientific article; zbMATH DE number 4063375
scientific article