Normality is upper join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property)
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Statement

Statement with symbols

Suppose $H$ is a subgroup of $G$, $I$ is a nonempty indexing set, and $K_{i},i\in I$ are subgroups of $G$ containing $H$, such that $H\triangleleft K_{i}$ (i.e., $H$ is a normal subgroup of $K_{i}$) for each $i\in I$. Then, $H$ is normal in the join of the $K_{i}$s.

Related facts

Related facts about normality

• Join lemma for normal subgroup of subgroup with normal subgroup of whole group
• Normality is not UL-join-closed
• Normality is strongly join-closed
• Normality is strongly intersection-closed
• Normality is UL-intersection-closed
• Normality satisfies intermediate subgroup condition
• Normality satisfies transfer condition

Related facts about upper join-closedness

The fact about normality generalizes to the following:

Left-inner right-monoidal implies upper join-closed: A subgroup property that has a function restriction expression with the left property being inner automorphisms and the right property being monoidal (closed under composition) is upper join-closed.

Other manifestations of the general fact include:

• Transitive normality is upper join-closed
• Central factor is upper join-closed

Here are some related properties that are not upper join-closed:

• Characteristicity is not upper join-closed
• Conjugacy-closedness is not upper join-closed
• Subnormality is not finite-upper join-closed, subnormality is not permuting upper join-closed
• 2-subnormality is not finite-upper join-closed, 2-subnormality is not permuting upper join-closed

Analogues and breakdowns of analogues in other algebraic structures

• Ideal property is upper join-closed for Lie rings: If $I$ is a subring of a Lie ring $L$ such that $I$ is an ideal in two subrings $A_{j}\leq L$, where $j\in J$, an indexing set, then $I$ is also an ideal in the Lie subring generated by all the $A_{j}$s.
• Ideal property is not upper join-closed for alternating rings
• Normality is not upper join-closed for algebra loops

Proof

Given: A group $G$, a subgroup $H$, a nonempty indexing set $I$, and a collection of subgroups $K_{i},i\in I$, such that $H$ is normal in $K_{i}$ for each $i\in I$.

To prove: $H$ is normal in the join of the $K_{i}$s.

Proof: Let $K$ be the join of the $K_{i}$s. For $g\in K$, we can write:

$g=g_{1}g_{2}g_{3}\dots g_{n}$

where $g_{j}\in K_{i_{j}}$ for some index element $i_{j}$. Thus, if $c_{g}$ denotes conjugation by $g$, we have:

$c_{g}=c_{g_{1}}\circ c_{g_{2}}\circ \dots \circ c_{g_{n}}$

Now, since $H$ is normal in $K_{i_{j}}$, each $c_{g_{j}}$ acts as an automorphism of $H$. Thus, their composite, namely $c_{g}$, is also an automorphism of $H$. In other words, $c_{g}(H)=H$ for every $g\in K$, showing that $H$ is normal in $K$.