Intermediate characteristicity is not transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., intermediately characteristic subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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An intermediately characteristic subgroup of an intermediately characteristic subgroup need not be intermediately characteristic.
Example of the dihedral group
Consider the dihedral group:D8, the dihedral group acting on a set of size four, i.e., the dihedral group with eight elements, given explicitly by the presentation:
The cyclic subgroup generated by is a subgroup of order four, and since all the elements outside it have order two, it is a characteristic subgroup. Being maximal, it is intermediately characteristic. Within this, the cyclic subgroup of order two generated by is again intermediately characteristic.
However, the cyclic subgroup of order two generated by is not intermediately characteristic in the whole group: it is isomorphic to the two-element subgroup generated by .