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From Groupprops

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

This article gives the statement and proof of a subgroup property satisfying a subgroup metaproperty.
View a complete list of subgroup metaproperty satisfactions OR View a complete list of subgroup metaproperty dissatisfactions

Contents 1 Statement 1.1 Verbal statement 1.2 Symbolic statement 2 Definitions used 2.1 Normal subgroup 2.2 Strongly intersection-closed 3 Generalizations 4 Related results 4.1 Results following from the same generalization 4.2 Similar results 5 Proof 6 Consequences

Statement

Verbal statement

An arbitrary (possibly empty) intersection of normal subgroups of a group is normal.

Note: The use of the word strongly is to allow the empty intersection as well. We can also say that normality is intersection-closed and also identity-true.

Symbolic statement

Let I be an indexing set and H_i be a family of normal subgroups of G indexed by I. Then, the intersection, over all i in I, of the normal subgroups H_i, is also a normal subgroup of G. In symbols:

Definitions used

Normal subgroup

A subgroup N of a group G is said to be normal, if given any inner automorphism σ of G (viz a map sending x to gxg ^{− 1}), we have σ(N) ⊆ N.

Strongly intersection-closed

A subgroup property is termed strongly intersection-closed if given any family of subgroups having the property, their intersection also has the property. Note that just saying that a subgroup property is intersection-closed simply means that given any nonempty family of subgroups with the property, the intersection also has the property.

Thus, the property of being strongly intersection-closed is the conjunction of the properties of being intersection-closed and identity-true, viz satisfied by the whole group as a subgroup of itself.

Generalizations

The general result (of which this can be viewed as a special case) is that any invariance property is strongly intersection-closed.

Here, an invariance property is the property of being invariant with respect to a certain collection of functions on the whole group. For normal subgroups, the collection of functions is the inner automorphisms.

Related results

Results following from the same generalization

• Characteristicity is strongly intersection-closed
• Fully characteristic satisfies strongly intersection-closed

Similar results

• Normality is strongly join-closed
• Characteristicity is strongly join-closed

Proof

Let H_i be a family of normal subgroups of G indexed by I. Suppose is the intersection. We need to show that . In other words, for any x in H and any inner automorphism σ of G, we need to show that .

Now we know that since x is in H, . By the normality of H_i, . Hence, .

Consequences

A consequence of normality being strongly intersection-closed is the fact that given any subgroup we can talk of the smallest normal subgroup containing that subgroup. This smallest normal subgroup is termed the normal closure.