Normality satisfies intermediate subgroup condition

This article gives the statement, and possibly proof, of a basic fact in group theory.
View a complete list of basic facts in group theory
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|
VIEW: Survey articles about this

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about normal subgroup |Get facts that use property satisfaction of normal subgroup | Get facts that use property satisfaction of normal subgroup|Get more facts about intermediate subgroup condition

Statement

Verbal statement

If a subgroup is normal in the whole group, it is also normal in every intermediate subgroup of the group containing it.

Statement with symbols

Let $H\leq K\leq G$ be groups such that $H\triangleleft G$ (viz., $H$ is normal in $G$). Then, $H$ is normal in $K$.

Property-theoretic statement

The subgroup property of being normal satisfies the Intermediate subgroup condition (?).

Related facts

Related metaproperties satisfied by normality

Here are some stronger metaproperties that normality satisfies:

• Normality satisfies transfer condition: If $H$ is normal in $G$ and $K\leq G$ is any subgroup, then $H\cap K$ is normal in $K$.
• Normality satisfies inverse image condition: If $\varphi :K\to G$ is a homomorphism and $H$ is normal in $G$, $\varphi ^{-1}(H)$ is normal in $K$.

Here are some other related metaproperties that normality satisfies:

• Normality satisfies image condition
• Normality is upper join-closed

General conditions to ensure intermediate subgroup condition

• Left-inner implies intermediate subgroup condition
• Left-extensibility-stable implies intermediate subgroup condition

Intermediate subgroup condition for related properties

Here are some other properties that satisfy the intermediate subgroup condition:

• Central factor satisfies intermediate subgroup condition
• Direct factor satisfies intermediate subgroup condition
• Subnormality satisfies intermediate subgroup condition

Here are some that don't:

• Characteristicity does not satisfy intermediate subgroup condition
• Full characteristicity does not satisfy intermediate subgroup condition

Analogues in other algebraic structures

• I-automorphism-invariance satisfies intermediate subgroup condition: An I-automorphism in a variety of algebras is an automorphism expressible by a formula that is always guaranteed to yield automorphisms. In the variety of groups, the I-automorphisms are precisely the inner automorphisms.
• Ideal property satisfies intermediate subalgebra condition: In any variety of algebras, an ideal of an algebra is also an ideal in every intermediate subalgebra containing it.
• Ideal property satisfies intermediate subring condition in Lie rings: In a Lie ring, any ideal is also an ideal in every intermediate Lie subring.

Proof

Hands-on proof

Given: $H\leq K\leq G$ such that $H\triangleleft G$

To prove: $H\triangleleft K$: for any $g\in K$, $gHg^{-1}=H$.

Proof: Pick any $g\in K$. Since $K\leq G$, $g\in G$. Further, since $H$ is normal in $G$ and $g\in G$, $gHg^{-1}=H$.

Proof in terms of inner automorphisms

This proof generalizes to the idea that I-automorphism-invariance satisfies intermediate subgroup condition over arbitrary varieties of algebras, and it also generalizes to results like left-inner implies intermediate subgroup condition and left-extensibility-stable implies intermediate subgroup condition.

The key idea here is that since inner automorphisms can be expressed by a formula that is guaranteed to yield an automorphism, any inner automorphism of a smaller subgroup extends to an inner automorphism of a bigger subgroup.

Given: $H\leq K\leq G$, such that $H$ is invariant under all inner automorphisms of $G$.

To prove: $H$ is invariant under all inner automorphisms of $K$.

Proof: Suppose $\sigma$ is an inner automorphism of $K$. Our goal is to show that $\sigma (H)\leq H$.

1. Since $\sigma$ is inner in $K$, there exists $g\in K$ such that $\sigma =c_{g}$. In other words, $\sigma (x)=gxg^{-1}$ for all $x\in H$.
2. Since $K\leq G$ and $g\in K$, we have $g\in G$.
3. The map $c_{g}:x\mapsto gxg^{-1}$ defines an inner automorphism $\sigma '$ of the whole group $G$, whose restriction to $K$ is $\sigma$.
4. Since $H$ is normal in $G$, $\sigma '(H)\leq H$.
5. Since the restriction of $\sigma '$ to $K$ is $\sigma$, and $H\leq K$, we get $\sigma (H)\leq H$.