Commutator of a group and a subgroup implies normal
From Groupprops
This article describes a computation relating the result of the commutator operator on two known subgroup properties or properties of subsets of groups: (i.e., improper subgroup and subgroup), to another known subgroup property (i.e., normal subgroup)
View a complete list of commutator computations
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subgroup realizable as the commutator of the whole group and a subgroup) must also satisfy the second subgroup property (i.e., normal subgroup)
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Get more facts about subgroup realizable as the commutator of the whole group and a subgroup|Get more facts about normal subgroup
Contents
Statement
Statement with symbols
Suppose is a group and
is any subgroup. Then, the commutator
, defined as:
is a normal subgroup of .
Related facts
Stronger facts
- Commutator of a group and a subset implies normal
- Subgroup normalizes its commutator with any subset
- Commutator of a group and a subgroup of its automorphism group is normal
- Commutator of two subgroups is normal in join
- Commutator of a normal subgroup and a subset implies 2-subnormal
- Commutator of a 2-subnormal subgroup and a subset implies 3-subnormal
- Normality is commutator-closed
- Characteristicity is commutator-closed
Breakdown for Lie rings
Facts used
- Subgroup normalizes its commutator with any subset: If
and
is a subset of
, then
normalizes the commutator
.
Proof
Given: A group , a subgroup
.
To prove: The subgroup is normal in
.
Proof: Apply fact (1) to and
. We get that
normalizes
. Hence,
is normal in
.