Intermediately normal-to-characteristic implies intermediately subnormal-to-normal
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., intermediately normal-to-characteristic subgroup) must also satisfy the second subgroup property (i.e., intermediately subnormal-to-normal 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
Property-theoretic statement
The subgroup property of being intermediately normal-to-characteristic implies the subgroup property of being intermediately subnormal-to-normal.
Definitions used
Intermediately subnormal-to-normal subgroup
Further information: Intermediately subnormal-to-normal subgroup
A subgroup of a group
is termed intermediately subnormal-to-normal if whenever
is a subgroup containing
such that
is 2-subnormal in
, then
is normal in
.
Intermediately normal-to-characteristic subgroup
Further information: Intermediately normal-to-characteristic subgroup
A subgroup of a group
is termed intermediately normal-to-characteristic if whenever
is a subgroup containing
such that
is normal in
,
is characteristic in
.
Facts used
Proof
Given: A subgroup of a group
, such that
is intermediately normal-to-characteristic in
.
To prove: For any subgroup of
containing
, such that
is 2-subnormal in
,
is normal in
.
Proof: Since is 2-subnormal in
, there exists a subgroup
such that
.
Now, is a subgroup of
, and
is normal in
. Since
is intermediately normal-to-characteristic in
,
is characteristic in
.
Thus, is characteristic in
, that in turn is normal in
. By fact (1),
is normal in
, completing the proof.