Characteristic subgroup

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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.[1][2] 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.

Contents

Definitions

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.