Characteristic subgroup

A subgroup {{math|H}} of a group {{math|G}} is called a characteristic subgroup if for every automorphism {{math|φ}} of {{math|G}}, one has {{math|φ(H) ≤ H}}; then write {{math|H char G}}.

It would be equivalent to require the stronger condition {{math|φ(H)}} = {{math|H}} for every automorphism {{math|φ}} of {{math|G}}, because {{math|φ⁻¹(H) ≤ H}} implies the reverse inclusion {{math|H ≤ φ(H)}}.

Given {{math|H char G}}, every automorphism of {{math|G}} induces an automorphism of the quotient group {{math|G/H}}, which yields a homomorphism {{math|Aut(G) → Aut(G/H)}}.

If {{math|G}} has a unique subgroup {{math|H}} of a given index, then {{math|H}} is characteristic in {{math|G}}.

{{main|Normal subgroup}}

A subgroup of {{math|H}} that is invariant under all inner automorphisms is called normal; also, an invariant subgroup.

:{{math|∀φ ∈ Inn(G)： φ(H) ≤ H}}

Since {{math|Inn(G) ⊆ 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 {{math|H}} be a nontrivial group, and let {{math|G}} be the direct product, {{math|H × H}}. 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, {{math|(x, y) → (y, x)}}, that switches the two factors.
• For a concrete example of this, let {{math|V}} be the Klein four-group (which is isomorphic to the direct product, $\backslash mathbb\{Z\}\_2\; \backslash times\; \backslash mathbb\{Z\}\_2$). Since this group is abelian, every subgroup is normal; but every permutation of the 3 non-identity elements is an automorphism of {{math|V}}, 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, {{math|T(e) {{=}} e, T(a) {{=}} b, T(b) {{=}} a, T(ab) {{=}} ab}}; then {{math|T(H)}} is not contained in {{math|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, {{math|{1, −1{{)}}}}, is characteristic, since it is the only subgroup of order 2.
• If {{math|n}} > 2 is even, the dihedral group of order {{math|2n}} 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.

A {{vanchor|strictly characteristic subgroup}}, or a {{vanchor|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 being strictly characteristic is equivalent to characteristic. This is not the case anymore for infinite groups.

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 {{math|G}} (and not just every automorphism):

:{{math|∀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.

{{cite book

| title = Group Theory

| first = W.R. | last = Scott

| pages = 45–46

| publisher = Dover

| year = 1987

| isbn = 0-486-65377-3

}}

{{cite book

| title = Combinatorial Group Theory

| first1 = Wilhelm | last1 = Magnus

| first2 = Abraham | last2 = Karrass

| first3 = Donald | last3 = Solitar

| publisher = Dover

| year = 2004

| pages = 74–85

| isbn = 0-486-43830-9

}}

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

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.

The property of being characteristic or fully characteristic is transitive; if {{math|H}} is a (fully) characteristic subgroup of {{math|K}}, and {{math|K}} is a (fully) characteristic subgroup of {{math|G}}, then {{math|H}} is a (fully) characteristic subgroup of {{math|G}}.

:{{math|H char K char G ⇒ H char G}}.

Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal.

:{{math|H char K ⊲ G ⇒ H ⊲ G}}

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 {{math|H char G}} and {{math|K}} is a subgroup of {{math|G}} containing {{math|H}}, then in general {{math|H}} is not necessarily characteristic in {{math|K}}.

:{{math|H char G, H < K < G ⇏ H char K}}

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) × $\backslash mathbb\{Z\}\; /\; 2\; \backslash mathbb\{Z\}$}}, has a homomorphism taking {{math|(π, y)}} to {{math|((1, 2){{sup|y}}, 0)}}, which takes the center, $1\; \backslash times\; \backslash mathbb\{Z\}\; /\; 2\; \backslash mathbb\{Z\}$, into a subgroup of {{math|Sym(3) × 1}}, 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

Consider the group {{math|G {{=}} S{{sub|3}} × $\backslash mathbb\{Z\}\_2$}} (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 {{math|G}} is isomorphic to its second factor $\backslash mathbb\{Z\}\_2$. Note that the first factor, {{math|S{{sub|3}}}}, contains subgroups isomorphic to $\backslash mathbb\{Z\}\_2$, for instance {{math|{e, (12)} }}; let be the morphism mapping $\backslash mathbb\{Z\}\_2$ onto the indicated subgroup. Then the composition of the projection of {{math|G}} onto its second factor $\backslash mathbb\{Z\}\_2$, followed by {{math|f}}, followed by the inclusion of {{math|S{{sub|3}}}} into {{math|G}} as its first factor, provides an endomorphism of {{math|G}} under which the image of the center, $\backslash mathbb\{Z\}\_2$, is not contained in the center, so here the center is not a fully characteristic subgroup of {{math|G}}.

Every subgroup of a cyclic group is characteristic.

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.

The identity component of a topological group is always a characteristic subgroup.

• Characteristically simple group

{{reflist}}

Category:Subgroup properties