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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)
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 upper join-closed subgroup property

Contents 1 Statement 1.1 Statement with symbols 2 Related facts 2.1 Related facts about normality 2.2 Related facts about upper join-closedness 2.3 Analogues and breakdowns of analogues in other algebraic structures 3 Proof

Statement

Statement with symbols

Suppose H is a subgroup of G, I is a nonempty indexing set, and are subgroups of G containing H, such that (i.e., H is a normal subgroup of K_i) for each . Then, H is normal in the join of the K_is.

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 , where , an indexing set, then I is also an ideal in the Lie subring generated by all the A_js.
• 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 , such that H is normal in K_i for each .

To prove: H is normal in the join of the K_is.

Proof: Let K be the join of the K_is. For , we can write:

where for some index element i_j. Thus, if c_g denotes conjugation by g, we have:

Now, since H is normal in , each 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 , showing that H is normal in K.