Characteristic subgroup

A subgroup of a group is called a characteristic subgroup if for every automorphism of , one has ; then write ''''''. It would be equivalent to require the stronger condition = for every automorphism of , because implies the reverse inclusion . == Basic properties ==

Given , every automorphism of induces an automorphism of the quotient group , which yields a homomorphism . If has a unique subgroup of a given index, then is characteristic in . == Related concepts ==

Normal subgroup A subgroup of that is invariant under all inner automorphisms is called normal; also, an invariant subgroup. : Since and a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal. However, not every normal subgroup is characteristic. Here are several examples: • Let be a nontrivial group, and let be the direct product, . Then the subgroups, {{math|{1} × H}} and {{math|H × {1}}, are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism, , that switches the two factors. • For a concrete example of this, let be the Klein four-group (which is isomorphic to the direct product, \mathbb{Z}_2 \times \mathbb{Z}_2). Since this group is abelian, every subgroup is normal; but every permutation of the 3 non-identity elements is an automorphism of , so the 3 subgroups of order 2 are not characteristic. Here {{math|V {e, a, b, ab} }}. Consider {{math|H {e, a}} and consider the automorphism, ; then is not contained in . • 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, {{math|{1, −1}}, is characteristic, since it is the only subgroup of order 2. • If > 2 is even, the dihedral group of order has 3 subgroups of index 2, 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. Strictly characteristic subgroup A ', or a ', is one which is invariant under surjective endomorphisms. For finite groups, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism; thus being strictly characteristic is equivalent to characteristic. This is not the case anymore for infinite groups. Fully characteristic 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 endomorphism of (and not just every automorphism): :. 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. Every endomorphism of induces an endomorphism of , which yields a map . Verbal subgroup An even stronger constraint is verbal subgroup, which is the image of a fully invariant subgroup of a free group under a homomorphism. More generally, any verbal subgroup is always fully characteristic. For any reduced free group, and, in particular, for any free group, the converse also holds: every fully characteristic subgroup is verbal. == Transitivity ==

The property of being characteristic or fully characteristic is transitive; if is a (fully) characteristic subgroup of , and is a (fully) characteristic subgroup of , then is a (fully) characteristic subgroup of . :. Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal. : Similarly, while being strictly characteristic (distinguished) is not transitive, it is true that every fully characteristic subgroup of a strictly characteristic subgroup is strictly characteristic. However, unlike normality, if and is a subgroup of containing , then in general is not necessarily characteristic in . : == Containments ==

Every subgroup that is fully characteristic is certainly strictly characteristic and characteristic; but a characteristic or even strictly characteristic subgroup need not be fully characteristic. The center of a group is always a strictly characteristic subgroup, but it is not always fully characteristic. For example, the finite group of order 12, {{math|Sym(3) × \mathbb{Z} / 2 \mathbb{Z}}}, has a homomorphism taking to , which takes the center, 1 \times \mathbb{Z} / 2 \mathbb{Z}, into a subgroup of , which meets the center only in the identity. The relationship amongst these subgroup properties can be expressed as: :Subgroup ⇐ Normal subgroup ⇐ Characteristic subgroup ⇐ Strictly characteristic subgroup ⇐ Fully characteristic subgroup ⇐ Verbal subgroup ==Examples==

Finite example Consider the group {{math|G S × \mathbb{Z}_2}} (the group of order 12 that is the direct product of the symmetric group of order 6 and a cyclic group of order 2). The center of is isomorphic to its second factor \mathbb{Z}_2. Note that the first factor, , contains subgroups isomorphic to \mathbb{Z}_2, for instance {{math|{e, (12)} }}; let f: \mathbb{Z}_2 be the morphism mapping \mathbb{Z}_2 onto the indicated subgroup. Then the composition of the projection of onto its second factor \mathbb{Z}_2, followed by , followed by the inclusion of into as its first factor, provides an endomorphism of under which the image of the center, \mathbb{Z}_2, is not contained in the center, so here the center is not a fully characteristic subgroup of . Cyclic groups Every subgroup of a cyclic group is characteristic. Subgroup functors The derived subgroup (or commutator subgroup) of a group is a verbal subgroup. The torsion subgroup of an abelian group is a fully invariant subgroup. Topological groups The identity component of a topological group is always a characteristic subgroup. ==See also==