BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 with symbols
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.