# Intersection of subgroups is subgroup

This article gives the statement, and possibly proof, of a basic fact in group theory.
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## Statement

### Verbal statement

The intersection of any arbitrary collection of subgroups of a group is again a subgroup.

### Symbolic statement

Let  be an arbitrary collection of subgroups of a group  indexed by . Then,  is again a subgroup of .

Note that if the collection  is empty, the intersection is defined to be the whole group. In this case, the intersection is clearly a subgroup. It should be noted that the case of an empty intersection is covered in the language of the general proof.

## Related facts

For examples, see the article Intersection of subgroups

### The related notion of join of subgroups

Given a collection of subgroups, their join is defined as the smallest subgroup containing all of them; equivalently, it is the intersection of all subgroups containing them.

This is closely related to the notion of the subgroup generated by a subset. The subgroup generated by a subset is the intersection of all subgroups containing that subset.

Notice that although the union of subgroups is not a subgroup, the fact that an intersection of subgroups is a subgroup tells us that there is a smallest subgroup containing any given collection of subgroups. This is analogous to the fact that the greatest lower bound property on a totally ordered set yields the least upper bound property.

### Other facts about intersections of subgroups

A subgroup property is termed:

• intersection-closed if the intersection of an arbitrary nonempty collection of subgroups with the property also has the property.
• finite-intersection-closed if the intersection of a finite nonempty collection of subgroups with the property also has the property.
• strongly intersection-closed if it is intersection-closed and also true for the whole group as a subgroup of itself. Thus, it is preserved on taking intersections of possibly empty collections.
• strongly finite-intersection-closed if it is finite-intersection-closed and also true for the whole group as a subgroup of itself.

There are some basic results of importance about intersection-closedness:

### Intersections of subsets other than subgroups

Subset property Proof that it is closed under arbitrary intersections
twisted subgroup intersection of twisted subgroups is twisted subgroup
1-closed subset intersection of 1-closed subsets is 1-closed subset
symmetric subset intersection of symmetric subsets is symmetric subset

### Analogues in other algebraic structures

For any variety of algebras, an intersection of subalgebras is a subalgebra. The proof is exactly the same as that for groups. In fact, the result holds in a slightly greater generality than varieties of algebras. For instance, an intersection of subfields is a subfield, although fields do not form a variety of algebras.

## Proof

Given: Let  be an arbitrary collection of subgroups of a group  indexed by  Let us denote  Here,  denotes the identity element of 

To prove: We need to show that  is a subgroup. In other words, we need to show the following:

1. 
2. If  then 
3. If  then 

Proof: Let's prove these one by one:

1. Since  for every  
2. Take . Then  for every  Since each  is a subgroup,  for each  Thus, 
3. Take  Then  for every  so  for every  Thus