Left transiter of normal is characteristic
Template:Left transiter computation
Statement
Statement with symbols
Let be a subgroup. The following are equivalent:
- is a Characteristic subgroup (?) of
- If is a group containing such that is a Normal subgroup (?) of , then is also a normal subgroup of .
Property-theoretic statement
The left transiter of the subgroup property of normality is the subgroup property of characteristicity. In other words:
- Characteristic Normal Normal
Every characteristic subgroup of a normal subgroup is normal (the here is for the composition operator).
- If is such that:
Normal Normal
Then Characteristic
Definitions used
Characteristic subgroup
- Hands-on definition: A subgroup of a group is termed characteristic in if for any automorphism of , .
- Definition using function restriction expression: A subgroup of a group is termed characteristic in if it has the following function restriction expression:
Automorphism Automorphism
In other words, every automorphism of restricts to an automorphism of .
Normal subgroup
- Hands-on definition: A subgroup of a group is termed normal in if for every , .
- Definition using function restriction expression: A subgroup of a group is termed normal in if it has the following function restriction expression:
Inner automorphism Automorphism
In other words, every inner automorphism of restricts to an automorphism of .
Related facts
Related left residual computations
All these use the fact that inner automorphism to automorphism is right tight for normality.
- Left residual of conjugate-permutable by normal is automorph-permutable
- Left residual of pronormal by normal is procharacteristic
- Left residual of weakly pronormal by normal is weakly procharacteristic
- Left residual of paranormal by normal is paracharacteristic
- Left residual of polynormal by normal is polycharacteristic
- Left residual of weakly normal by normal is weakly characteristic
Some examples with a somewhat different flavor:
- Characteristic implies left-transitively 2-subnormal, Characteristic implies left-transitively fixed-depth subnormal
- Subgroup-coprime automorphism-invariant implies left-transitively 2-subnormal
- Complemented normal is not transitive
- Left-transitively complemented normal implies characteristic, Characteristic not implies left-transitively complemented normal
- Left-transitively permutable implies characteristic
- Normal upper-hook fully normalized implies characteristic: If are such that is normal in and is fully normalized in , then is characteristic in .
- Characteristically simple and normal fully normalized implies minimal normal
Facts used
- Characteristic of normal implies normal (used for the more direct part of the proof).
- Inner automorphism to automorphism is right tight for normality
Proof
Hands-on proof
Given: A subgroup of a group .
To prove: The following are equivalent:
- is characteristic in .
- For every group in which is normal, is normal in .
Proof: (1) implies (2) follows from fact (1). We now prove (2) implies (1).
For the group , let denote the holomorph of . is the semidirect product of with . Observe that every automorphism of lifts to an inner automorphism of , and further, that is a normal subgroup of .
Now let be a subgroup of with the property that whenever is normal in a group , so is . We will show that is characteristic in .
Take . Clearly is normal in . Now, let . Clearly, there is an inner automorphism of whose restriction to is (namely, conjugation by the element of that is ). But is normal in , so every inner automorphism of must leave invariant, and hence must leave invariant. This shows that is invariant under all automorphisms of , and hence, is a characteristic subgroup.
Property-theoretic proof
By fact (2), the function restriction expression:
Inner automorphism Automorphism
is a right tight function restriction expression for normality. In other words, for every automorphism of a group, there is a bigger group in which it is normal, such that the automorphism extends to an inner automorphism in that bigger group. Combining this with fact (3), we see that the left transiter of normality is the property:
Automorphism Automorphism
which is indeed the property of being a characteristic subgroup.