Isomorph-freeness is not transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., isomorph-free subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about isomorph-free subgroup|Get more facts about transitive subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., isomorph-containing subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about isomorph-containing subgroup|Get more facts about transitive subgroup property|

Statement

Verbal statement

An isomorph-free subgroup of an isomorph-free subgroup need not be isomorph-free.

This also shows that an isomorph-containing subgroup of an isomorph-containing subgroup need not be isomorph-containing. Note that inside a finite group, the notions of isomorph-free and isomorph-containing are equivalent.

Proof

Example of the dihedral group

Further information: dihedral group:D8, subgroup structure of dihedral group:D8, cyclic maximal subgroup of dihedral group:D8, center of dihedral group:D8

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:

$\langle a,x\mid a^{4}=x^{2}=e,xax^{-1}=a^{-1}\rangle$.

The cyclic subgroup generated by $a$ is a subgroup of order four, and since all the elements outside it have order two, it is an isomorph-free subgroup. Within this, the cyclic subgroup of order two generated by $a^{2}$ is an isomorph-free subgroup.

However, the cyclic subgroup of order two generated by $a^{2}$ is not isomorph-free in the whole group: it is isomorphic to the two-element subgroup generated by $x$.