Cohopfian Groups And Accessible Group Classes

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 X such that G is minimal non-X. If X is any group class, the subclass X0 consisting of all groups that are isomorphic to proper subgroups of locally graded minimal non-X groups is often much smaller than X. Similarly, if Xprop is the class of all groups isomorphic to proper subgroups of X-groups, the class X of all locally graded minimal non-Xprop groups may contain many groups which are not in X. This paper investigates the relation between the classes X, X0 and X.