Every subgroup is contracharacteristic in its normal closure
From Groupprops
This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Contracharacteristic subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., Subgroup (?))
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This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
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Contents
Statement
Any subgroup of a group is a contracharacteristic subgroup of its normal closure. In particular, it occurs as a contracharacteristic subgroup of a normal subgroup.
Definitions used
Contracharacteristic subgroup
Further information: Contracharacteristic subgroup
A subgroup of a group is termed contracharacteristic if it is not contained in any proper characteristic subgroup.
Related facts
- Characteristic of normal implies normal
- Equivalence of definitions of subgroup of Abelian normal subgroup
Facts used
Proof
Hands-on proof
Given: A subgroup ,
is the normal closure of
in
.
To prove: If is a characteristic subgroup of
containing
, then
.
Proof:
- By fact (1), we see that since
is characteristic in
and
is normal in
, we obtain that
is normal in
.
- Thus,
is a normal subgroup of
containing
. By definition of normal closure, we get that
. Since
, we get
.