Perfect core is homomorph-containing
This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., perfect core) always satisfies a particular subgroup property (i.e., homomorph-containing subgroup)}
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Suppose is a group and is the perfect core of : it is the unique largest perfect subgroup of . Then, is a homomorph-containing subgroup of : given any homomorphism , we have .
- Perfectness is quotient-closed: The image of a perfect group under a surjective homomorphism is perfect.
Given: A group with perfect core and a homomorphism .
To prove: .
Proof: is perfect by fact (1). By the definition of perfect core, any perfect subgroup of is contained in . Thus, .