Normal and self-centralizing implies normality-large
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 normal subgroup) must also satisfy the second subgroup property (i.e., normality-large normal subgroup)
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Given: A group , a normal subgroup , such that .
To prove: If is a nontrivial normal subgroup of such that is trivial, then is trivial.
Proof: Consider the commutator . This is contained both in and in , since they are both normal. Hence it is contained in , which is trivial. Thus, every element of commutes with every element of , so . Since is self-centralizing, we get . Since is trivial, we get that is trivial.