Normalitylargeness is transitive
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Contents
Statement
Propertytheoretic statement
The subgroup property of being a normalitylarge subgroup satisfies the subgroup metaproperty of being transitive.
Verbal statement
Any normalitylarge subgroup of a normalitylarge subgroup is normalitylarge.
Statement with symbols
Suppose is a normalitylarge subgroup of and is a normalitylarge subgroup of . Then, is a normalitylarge subgroup of .
Proof
Handson proof
Given: A group , a normalitylarge subgroup of , and a subgroup of that is normalitylarge in
To prove: is a normalitylarge subgroup of
Proof: We need to show that if is a nontrivial normal subgroup of , then is nontrivial.
First, observe that since is normalitylarge in , is a nontrivial subgroup of . Now, since is normalitylarge in , is a nontrivial subgroup of . Since is contained in , , so is nontrivial.