Self-centralizing and minimal normal implies strictly characteristic
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., self-centralizing minimal normal subgroup) must also satisfy the second subgroup property (i.e., strictly characteristic subgroup)
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Any Minimal normal subgroup (?) that is also self-centralizing (i.e., contains its centralizer in the whole group) is strictly characteristic: it is invariant under any surjective endomorphism of the group.
Minimal normal subgroup
Further information: minimal normal subgroup
A minimal normal subgroup of a group is a nontrivial normal subgroup that does not properly contain any other nontrivial normal subgroup.
Further information: self-centralizing subgroup
A subgroup of a group is termed self-centralizing if it contains its centralizer in the whole group.
Strictly characteristic subgroup
Further information: strictly characteristic subgroup
A subgroup of a group is termed strictly characteristic if for every surjective endomorphism of , .
- Self-centralizing and minimal normal implies monolith
- Self-centralizing and minimal normal implies characteristic
- Self-centralizing and minimal normal implies monolith: In other words, a self-centralizing minimal normal subgroup is contained in every nontrivial normal subgroup.
- Monolith is strictly characteristic: A subgroup contained in every nontrivial normal subgroup is strictly characteristic.
Given: A group , a minimal normal subgroup such that . A surjective endomorphism of .
To prove: .
Proof: Consider the subgroup . This is normal by fact (2), so by fact (1) either is trivial or . Since is surjective and is nontrivial, cannot be trivial. Thus, . This forces that , as desired.