Editing Characteristic subgroup (section)

=== Fully characteristic subgroup{{anchor|Fully invariant subgroup}} ===
For an even stronger constraint, a ''fully characteristic subgroup'' (also, ''fully invariant subgroup'') of a group ''G'', is a subgroup ''H'' ≤ ''G'' that is invariant under every [[group homomorphism#Types|endomorphism]] of {{math|''G''}} (and not just every automorphism):
:{{math|∀φ ∈ End(''G'')： φ(''H'') ≤ ''H''}}.

Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups. The [[commutator subgroup]] of a group is always a fully characteristic subgroup.<ref>
{{cite book
 | title = Group Theory
 | first = W.R. | last = Scott
 | pages = 45–46
 | publisher = Dover
 | year = 1987
 | isbn = 0-486-65377-3
}}</ref><ref>
{{cite book
 | title = Combinatorial Group Theory
 | first1 = Wilhelm | last1 = Magnus
 | first2 = Abraham | last2 = Karrass
 | first3 = Donald  | last3 = Solitar
 | publisher = Dover
 | year = 2004
 | pages = 74–85
 | isbn = 0-486-43830-9
}}</ref>

Every endomorphism of {{math|''G''}} induces an endomorphism of {{math|''G/H''}}, which yields a map {{math|End(''G'') → End(''G''/''H'')}}.