In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group. Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal subgroup is characteristic. Examples of characteristic subgroups include the commutator subgroup and the center of a group.
A characteristic subgroup of a group G is a subgroup H that is invariant under each automorphism of G. That is,
for every automorphism φ of G (where φ(H) denotes the image of H under φ).
The statement “H is a characteristic subgroup of G” is written
Characteristic vs. normal
If G is a group, and g is a fixed element of G, then the conjugation map
is an automorphism of G (known as an inner automorphism). A subgroup of G that is invariant under all inner automorphisms is called normal. Since a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal.
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