Full invariance does not satisfy intermediate subgroup condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., fully invariant subgroup) not satisfying a subgroup metaproperty (i.e., intermediate subgroup condition).
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Statement with symbols
It is possible to have groups such that is a fully invariant subgroup of but is not a fully invariant subgroup of .
Intermediate subgroup condition
- Homomorph-containment satisfies intermediate subgroup condition
- Homomorph-containing implies intermediately fully invariant
- Characteristicity does not satisfy intermediate subgroup condition
Abelian example of prime-cube order
Let be any prime. Consider the group:
Let be the set of powers in and be the set of elements of order or . Then , and:
- is fully invariant in : This is on account of it being an agemo subgroup -- the image of a power under an endomorphism continues to be a power.
- is not fully invariant in : In fact, is where , so is an elementary abelian group of order . In particular, has an automorphism interchanging the coordinates, and is not invariant under this automorphism.
Example of the dihedral group
Let be the dihedral group of order eight:
Let and . Then: