Normality-largeness is transitive

From Groupprops
Jump to: navigation, search
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
|
Property "Page" (as page type) with input value "{{{property}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
Property "Page" (as page type) with input value "{{{metaproperty}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.


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

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]