Perfect core is homomorph-containing

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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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Statement

Suppose G is a group and H is the perfect core of G: it is the unique largest perfect subgroup of G. Then, H is a homomorph-containing subgroup of G: given any homomorphism \varphi:H \to G, we have \varphi(H) \le H.

Related facts

Facts used

  1. Perfectness is quotient-closed: The image of a perfect group under a surjective homomorphism is perfect.

Proof

Given: A group G with perfect core H and a homomorphism \varphi:H \to G.

To prove: \varphi(H) \le H.

Proof: \varphi(H) is perfect by fact (1). By the definition of perfect core, any perfect subgroup of G is contained in H. Thus, \varphi(H) \le H.