Normality-largeness is transitive

This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Statement

Property-theoretic statement

The subgroup property of being a normality-large subgroup satisfies the subgroup metaproperty of being transitive.

Verbal statement

Any normality-large subgroup of a normality-large subgroup is normality-large.

Statement with symbols

Suppose $H$ is a normality-large subgroup of $K$ and $K$ is a normality-large subgroup of $G$ . Then, $H$ is a normality-large subgroup of $G$ .

Proof

Hands-on proof

Given: A group $G$ , a normality-large subgroup $K$ of $G$ , and a subgroup $H$ of $K$ that is normality-large in $K$

To prove: $H$ is a normality-large subgroup of $G$

Proof: We need to show that if $N$ is a nontrivial normal subgroup of $G$ , then $H \cap N$ is nontrivial.

First, observe that since $K$ is normality-large in $G$ , $K \cap N$ is a nontrivial subgroup of $K$ . Now, since $H$ is normality-large in $K$ , $H \cap (K \cap N)$ is a nontrivial subgroup of $H$ . Since $H$ is contained in $K$ , $H \cap K \cap N = H \cap N$ , so $H \cap N$ is nontrivial.

Using the intersection restriction formalism

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