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Cohopfian groups and accessible group classes. (English) Zbl 07390399

Summary: A group \(G\) is said to be cohopfian if it is neither trivial nor isomorphic to any of its proper subgroups, and this property is equivalent to the existence of a suitable group class \(\mathfrak{X}\) such that \(G\) is minimal non-\( \mathfrak{X} \). If \(\mathfrak{X}\) is any group class, the subclass \(\mathfrak{X}^\circ\) consisting of all groups that are isomorphic to proper subgroups of locally graded minimal non-\( \mathfrak{X}\) groups is often much smaller than \(\mathfrak{X} \). Similarly, if \(\mathfrak{X}^{\operatorname{prop}}\) is the class of all groups isomorphic to proper subgroups of \(\mathfrak{X} \)-groups, the class \(\overline{\mathfrak{X}}\) of all locally graded minimal non-\( \mathfrak{X}^{\operatorname{prop}}\) groups may contain many groups which are not in \(\mathfrak{X} \). This paper investigates the relation between the classes \(\mathfrak{X}, \mathfrak{X}^\circ\) and \(\overline{\mathfrak{X}} \).

MSC:

20E34 General structure theorems for groups
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