Normality is strongly intersection-closed

From Groupprops

Jump to: navigation, search
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 possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., strongly intersection-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 more facts about strongly intersection-closed subgroup property

Contents

Statement

Verbal statement

An arbitrary (possibly empty) intersection of subgroups 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 Hi be a family of normal subgroups of G indexed by I. Then, the intersection, over all i in I, of the normal subgroups Hi, is also a normal subgroup of G. In symbols:

\bigcap_{i \in I} H_i \triangleleft G

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

Similar results

Proof

Let Hi be a family of normal subgroups of G indexed by I. Suppose H = \bigcap_{i \in I} H_i is the intersection. We need to show that H \triangleleft G. In other words, for any x in H and any inner automorphism σ of G, we need to show that \sigma(x) \in H.

Now we know that since x is in H, x \in H_i \forall i \in I. By the normality of Hi, \sigma(x) \in H_i \forall i \in I. Hence, \sigma(x) \in \bigcap_{i \in I} H_i = H.

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.

References

Textbook references

Facts about Normality is strongly intersection-closedRDF feed
Fact aboutNormal subgroup  +, Strongly intersection-closed subgroup property  +, and Intersection of subgroups  +
Referenced inDummitFoote (?, ?, ?)  +, Herstein (?, ?, ?)  +, and RobinsonAA (?, ?, ?)  +
Stated inDummitFoote (?, ?, ?)  +, Herstein (?, ?, ?)  +, and RobinsonAA (?, ?, ?)  +
Retrieved from "http://groupprops.subwiki.org/wiki/Normality_is_strongly_intersection-closed"
Personal tools