Characteristic Subgroup - Characteristic Vs. Normal

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.

Not every normal subgroup is characteristic. Here are several examples:

• Let H be a group, and let G be the direct product H × H. Then the subgroups {1} × H and H × {1} are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism (x, y) → (y, x) that switches the two factors.
• For a concrete example of this, let V be the Klein four-group (which is isomorphic to the direct product Z₂ × Z₂). Since this group is abelian, every subgroup is normal; but every permutation of the three non-identity elements is an automorphism of V, so the three subgroups of order 2 are not characteristic.Here Consider H={e,a} and consider the automorphism .Then T(H) is not contained in H.
• In the quaternion group of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup {1, −1} is characteristic, since it is the only subgroup of order 2.

Note: If H is the unique subgroup of a group G, then H is characteristic in G.

• If n is even, the dihedral group of order 2n has three subgroups of index two, all of which are normal. One of these is the cyclic subgroup, which is characteristic. The other two subgroups are dihedral; these are permuted by an outer automorphism of the parent group, and are therefore not characteristic.
• "Normality" is not transitive, but Characteristic has a transitive property, namely if H Char K and K normal in G then H normal in G.