Normality is strongly joinclosed
This article gives the statement, and possibly proof, of a basic fact in group theory.
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This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Statement
Propertytheoretic statement
The subgroup property of being a Normal subgroup (?) satisfies the subgroup metaproperty of being strongly joinclosed.
Note: The use of the word strongly is to allow the empty collection as well. We can also say that normality is joinclosed and also trivially true.
Verbal statement
The join of an arbitrary (possibly empty) collection of normal subgroups of a group is normal. In other words, the subgroup generated by a family of normal subgroups is normal.
Statement with symbols
Let be an indexing set and be a family of normal subgroups of indexed by . Then, the subgroup generated by all the , is also a normal subgroup of .
Note that, since any normal subgroup is permutable, the join of a finitely many normal subgroups is equal to their product. Thus, we obtain that if are two normal subgroups of a group , so is the product of subgroups .
Definitions used
Normal subgroup
Further information: Normal subgroup
A subgroup of a group is said to be normal, if given any inner automorphism of (i.e., a map sending for some ), we have . In other words, for any , we have .
Strongly joinclosed
A subgroup property is termed strongly joinclosed if given any family of subgroups having the property, their join (viz., the subgroup generated by them) also has the property. Note that just saying that a subgroup property is joinclosed simply means that given any nonempty family of subgroups with the property, the join also has the property.
Thus, the property of being strongly joinclosed is the conjunction of the properties of being joinclosed and trivially true, viz., satisfied by the trivial subgroup.
Related facts
Related facts about normality
 Join lemma for normal subgroup of subgroup with normal subgroup of whole group: If is a normal subgroup of and is a normal subgroup of a subgroup of , then is normal in . Note that since is normal, this can be rewritten as: is normal in .
 Normality is strongly intersectionclosed: An arbitrary intersection of normal subgroups of a group is a normal subgroup.
 Normality is upper joinclosed: If and is a nonempty collection of subgroups containing such that is normal in each , is also normal in the join of the s.
 Normality is strongly ULintersectionclosed
Generalizations
The general result (of which this can be viewed as a special case) is that any endoinvariance property is strongly joinclosed. This in turn follows from the fact that homomorphisms commute with joins.
Here, an invariance property is the property of being invariant with respect to a certain collection of functions on the whole group. An endoinvariance property is an invariance property with respect to a collection of functions that are all endomorphisms. For normal subgroups, the collection of functions is the inner automorphisms.
Other instances of this generalization are:
Property  Endoinvariance property with respect to ...  Proof that it is strongly joinclosed 

Characteristic subgroup  automorphisms  Characteristicity is strongly joinclosed 
Fully invariant subgroup  endomorphisms  Full invariance is strongly joinclosed 
Strictly characteristic subgroup  surjective endomorphisms  Strict characteristicity is strongly joinclosed 
Injective endomorphisminvariant subgroup  injective endomorphisms  Injective endomorphisminvariance is strongly joinclosed 
Many subgroup properties closely related to normality fail to be joinclosed. For instance:
 Subnormality is not finitejoinclosed: A join of two subnormal subgroups of a group need not be subnormal.
 2subnormality is not finitejoinclosed: A join of two 2subnormal subgroups of a group need not be 2subnormal.
 Pronormality is not finitejoinclosed
On the other hand, some properties closely related to normality continue to be joinclosed:
Analogues in other algebraic structures
 Ideal property is strongly joinclosed: In a Lie ring, the Lie subring generated by a collection of ideals is again an ideal.
Variety of algebras perspective
 Variety of groups is congruencepermutable: The fact that a join of normal subgroups is normal, combined with the fact that the join of normal subgroups equals their product, is closely related to the fact that the variety of groups is congruencepermutable.
Proof
Note that for vacuous reasons, the proofs given here also work when is empty  for instance, in the first proof, the product described below has length zero but the arguments hold formally. However, it may be more clear to handle the case of the empty join separately by showing that trivial subgroup is normal.
Proof using conjugation definition
Given: A group with subgroups whose join is a subgroup of . Further, each is normal in .
To prove: For any and , .
Proof: We pick and .
 We can write where and . This is by the definition of join.
 We then have that . (This is computationally clear by associativity, but also follows from the conceptual idea that conjugations are automorphisms, i.e., group acts as automorphisms by conjugation).
 Each is in , because is normal in .
 From the previous two steps, is a product of elements, each of which is in for some and hence in . Thus, .
This completes the proof.
Proof using commutator definition
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Proof using coset definition
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]References
Textbook references
 Abstract Algebra by David S. Dummit and Richard M. Foote, 10digit ISBN 0471433349, 13digit ISBN 9780471433347, ^{More info}, Page 88, Exercise 23
 Topics in Algebra by I. N. Herstein, ^{More info}, Page 53, Problem 11 (this asks for a slight variant: in this problem we're taking the product, rather than the join of the subgroups. However, because normal implies permutable, the product of two normal subgroups is the same as their join)