Socle is strictly characteristic

Statement
The socle of a group (defined as the subgroup generated by all minimal normal subgroups) is a strictly characteristic subgroup: any surjective endomorphism of the whole group sends the socle to within itself.

Related facts

 * Monolith is strictly characteristic: This is a special case where there is a unique minimal normal subgroup, and this is contained in every nontrivial normal subgroup.
 * Baer norm is strictly characteristic
 * Fitting subgroup is strictly characteristic
 * Solvable core is strictly characteristic
 * Center is strictly characteristic

Facts used

 * 1) uses::Socle is normality-preserving endomorphism-invariant
 * 2) uses::Normality-preserving endomorphism-invariant implies strictly characteristic

Proof
The proof follows directly from facts (1) and (2).