Characteristic subgroup

Normal subgroup

A subgroup H {\displaystyle H} of G {\displaystyle G} that is invariant under all inner automorphisms is called normal; sometimes called an invariant subgroup.

Since Inn ⁡ ( G ) ⊆ Aut ⁡ ( G ) {\displaystyle \operatorname {Inn} (G)\subseteq \operatorname {Aut} (G)} 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 H {\displaystyle H} be a nontrivial group, and let G = H × H {\displaystyle G=H\times H} be the direct product. Then the subgroups { 1 } × H {\displaystyle \{1\}\times H} and H × { 1 } {\displaystyle H\times \{1\}} are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism ( x , y ) ↦ ( y , x ) {\displaystyle (x,y)\mapsto (y,x)} , that switches the two factors.

• • For a concrete example of this, let V ≅ Z 2 × Z 2 {\displaystyle V\cong \mathbb {Z} _{2}\times \mathbb {Z} _{2}} be the Klein four-group. Since this group is abelian, every subgroup is normal; however, every permutation of the 3 {\displaystyle 3} non-identity elements is an automorphism of V {\displaystyle V} , so the 3 {\displaystyle 3} subgroups of order 2 {\displaystyle 2} are not characteristic. Formally, V = { e , a , b , a b } {\displaystyle V=\{e,a,b,ab\}} . If H = { e , a } {\displaystyle H=\{e,a\}} for example and T {\displaystyle T} is the automorphism switching a {\displaystyle a} and b {\displaystyle b} , then T ( H ) {\displaystyle T(H)} is not contained in H {\displaystyle 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 } {\displaystyle \{1,-1\}} , is characteristic, since it is the only subgroup of order 2.

• If n > 2 {\displaystyle n>2} is even, the dihedral group of order 2 n {\displaystyle 2n} has 3 {\displaystyle 3} subgroups of index 2 {\displaystyle 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 strictly characteristic subgroup, or a distinguished subgroup, 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, strictly characteristic is equivalent to characteristic. This is not the case for infinite groups.

Fully characteristic subgroup

For an even stronger constraint, a fully characteristic subgroup (also, fully invariant subgroup) of a group G {\displaystyle G} , is a subgroup H ≤ G that is invariant under every endomorphism of G {\displaystyle G} (and not just every automorphism):

∀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.^[3][4]

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

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.

Finite example

Consider the group G = S 3 × Z 2 {\displaystyle G=S_{3}\times \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 G {\displaystyle G} is isomorphic to its second factor Z 2 {\displaystyle \mathbb {Z} _{2}} . Note that the first factor, S 3 {\displaystyle S_{3}} , contains subgroups isomorphic to Z 2 {\displaystyle \mathbb {Z} _{2}} , for instance e , ( 12 ) {\displaystyle {e,(12)}} ; let f : Z 2 → S 3 {\displaystyle f:\mathbb {Z} _{2}\rightarrow {\text{S}}_{3}} be the morphism mapping Z 2 {\displaystyle \mathbb {Z} _{2}} onto the indicated subgroup. Then the composition of the projection of G {\displaystyle G} onto its second factor Z 2 {\displaystyle \mathbb {Z} _{2}} , followed by f {\displaystyle f} , followed by the inclusion of S 3 {\displaystyle S_{3}} into G {\displaystyle G} as its first factor, provides an endomorphism of G {\displaystyle G} under which the image of the center, Z 2 {\displaystyle \mathbb {Z} _{2}} , is not contained in the center, so here the center is not a fully characteristic subgroup of G {\displaystyle G} .

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.