# Isomorph-automorphic subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

### Symbol-free definition

A subgroup of a group is termed **isomorph-automorphic** if given any other isomorphic subgroup, there is an automorphism of the whole group, mapping the subgroup isomorphically to the other one.

### Definition with symbols

A subgroup of a group is termed **isomorph-automorphic** if whenever there exists a subgroup of such that and are isomorphic groups, there exists an automorphism of such that .

## Examples

In a finite cyclic group, every subgroup is isomorph-automorphic (in fact, every subgroup is isomorph-free: no two subgroups are isomorphic).

Similarly, in a finite elementary Abelian group, every subgroup is isomorph-automorphic. That's because given any two subspaces of a finite-dimensional vector space that have the same dimension, there is an automorphism of the whole space taking one to the other.

By Sylow's theorem, every Sylow subgroup is isomorph-automorphic.

On the other hand, many subgroups are not isomorph-automorphic:

- In , the group of integers, any nontrivial subgroup is of the form , , hence is isomorphic to . However, there is clearly no automorphism of mapping it to a proper subgroup.
- Any infinite-dimensional vector space is isomorphic to a subspace of codimension one, but there is no automorphism mapping the whole space to such a subspace.
- In the direct product , the direct factor is isomorphic to the subgroup of multiples of 2 in the direct factor . However, there is no automorphism taking the first to the second.
- In the dihedral group of order eight, the center is of order two, and any subgroup generated by a reflection has order two, so they are isomorphic. However, there is no automorphism taking the center to a subgroup generated by a reflection.