Normality-largeness is transitive
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Any normality-large subgroup of a normality-large subgroup is normality-large.
Statement with symbols
Suppose is a normality-large subgroup of and is a normality-large subgroup of . Then, is a normality-large subgroup of .
Given: A group , a normality-large subgroup of , and a subgroup of that is normality-large in
To prove: is a normality-large subgroup of
Proof: We need to show that if is a nontrivial normal subgroup of , then is nontrivial.
First, observe that since is normality-large in , is a nontrivial subgroup of . Now, since is normality-large in , is a nontrivial subgroup of . Since is contained in , , so is nontrivial.