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

Statement with symbols

It is possible to have groups $H\leq K\leq G$ such that $H$ is a fully invariant subgroup of $G$ but $H$ is not a fully invariant subgroup of $K$.

Related facts

Intermediate subgroup condition

• Homomorph-containment satisfies intermediate subgroup condition
• Homomorph-containing implies intermediately fully invariant
• Characteristicity does not satisfy intermediate subgroup condition

Other related facts

• Full invariance does not satisfy image condition

Proof

Abelian example of prime-cube order

Let $p$ be any prime. Consider the group:

$G:=\mathbb {Z} /p\mathbb {Z} \times \mathbb {Z} /p^{2}\mathbb {Z}$.

Let $H=\mho ^{1}(G)=\{(0,pa)\}$ be the set of $p^{th}$ powers in $G$ and $K=\Omega _{1}(G)=\{(a,pb\}$ be the set of elements of order $1$ or $p$. Then $H\leq K\leq G$, and:

• $H$ is fully invariant in $G$: This is on account of it being an agemo subgroup -- the image of a $p^{th}$ power under an endomorphism continues to be a $p^{th}$ power.
• $H$ is not fully invariant in $K$: In fact, $K$ is $L\times H$ where $L=\{(a,0)\}$, so $K$ is an elementary abelian group of order $p^{2}$. In particular, $K$ has an automorphism interchanging the coordinates, and $H$ is not invariant under this automorphism.

Example of the dihedral group

Further information: dihedral group:D8, subgroup structure of dihedral group:D8

Let $G$ be the dihedral group of order eight:

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

Let $H=\langle a^{2}\rangle$ and $K=\langle a^{2},x\rangle$. Then:

• $H$ is fully invariant in $G$: In fact, $H=\mho ^{1}(G)$, and also $H$ is the commutator subgroup of $G$, so $H$ is fully invariant in $G$.
• $H$ is not fully invariant in $K$: In fact, $K=H\times L$ where $L=\langle x\rangle$, so $K$ is a Klein four-group and it admits an automorphism interchanging $H$ and $L$.