Solvable radical is normal-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., solvable radical) always satisfies a particular subgroup property (i.e., normal-homomorph-containing subgroup)}
View subgroup property satisfactions for subgroup-defining functions | View subgroup property dissatisfactions for subgroup-defining functions

Statement

Suppose H is the solvable radical of a group G, i.e., the unique largest solvable normal subgroup of G. (Note: The solvable radical may not exist for every group. It does, however, exist for finite groups, virtually solvable groups, and slender groups).

Then, H is a normal-homomorph-containing subgroup of G. In other words, for any homomorphism f:H \to G such that f(H) is a normal subgroup of G, f(H) \le H.

Related facts

Stronger facts: stronger subgroup properties satisfied

Property Meaning Proof of satisfaction by solvable radical Why it's stronger than normal-homomorph-containing
Normal-subhomomorph-containing subgroup contains any normal subgroup that is a homomorphic image of a subgroup solvable radical is normal-subhomomorph-containing obvious

Weaker facts: weaker subgroup properties satisfied

Property Meaning Proof of satisfaction by solvable radical Why it's weaker than normal-homomorph-containing
weakly normal-homomorph-containing subgroup invariant under homomorphisms from it to whole group that send normal subgroups to normal subgroups normal-homomorph-containing implies homomorph-containing
Strictly characteristic subgroup invariant under all surjective endomorphisms solvable radical is strictly characteristic normal-homomorph-containing implies strictly characteristic
Characteristic subgroup invariant under all automorphisms solvable radical is characteristic (via strictly characteristic)
Normal subgroup invariant under all inner automorphisms solvable radical is normal (via characteristic)