View source for Characteristic subgroup

{{Short description|Subgroup mapped to itself under every automorphism of the parent group}}
In [[Mathematics|mathematics]], particularly in the area of [[Abstract algebra|abstract algebra]] known as [[Group theory|group theory]], a '''characteristic subgroup''' is a [[Subgroup|subgroup]] that is mapped to itself by every [[Automorphism|automorphism]] of the parent [[Group (mathematics)|group]].<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher={{wipe|John Wiley & Sons}} | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge |  title=Algebra | publisher=[[Company:Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref>  Because every [[Conjugacy class#Properties|conjugation map]] is an [[Inner automorphism|inner automorphism]], every characteristic subgroup is [[Normal subgroup|normal]]; though the converse is not guaranteed.  Examples of characteristic subgroups include the [[Commutator subgroup|commutator subgroup]] and the center of a group.

== Definition ==
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<sup>−1</sup>(''H'') ≤ ''H''}} implies the reverse inclusion {{math|''H'' ≤ φ(''H'')}}.

== Basic properties ==

Given {{math|''H'' char ''G''}}, every automorphism of {{math|''G''}} induces an automorphism of the [[Quotient group|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''}}.

== Related concepts ==

=== Normal subgroup ===
{{main|Normal subgroup}}
A subgroup of {{math|''H''}} that is invariant under all inner automorphisms is called [[Normal subgroup|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 of groups|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 [[Group isomorphism|isomorphic]] to the direct product, <math>\mathbb{Z}_2 \times \mathbb{Z}_2</math>).  Since this group is [[Abelian group|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|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''}} is even, the [[Dihedral group|dihedral group]] of order {{math|2''n''}} has 3 subgroups of [[Index of a subgroup|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 group|outer automorphism]] of the parent group, and are therefore not characteristic.

=== Strictly characteristic subgroup{{anchor|Strictly invariant subgroup}} ===
A ''{{vanchor|strictly characteristic subgroup}}'', or a ''{{vanchor|distinguished subgroup}}'', which is invariant under surjective [[Endomorphism|endomorphism]]s. For [[Finite group|finite group]]s, 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{{anchor|Fully invariant subgroup}} ===
For an even stronger constraint, a ''fully characteristic subgroup'' (also, ''fully invariant subgroup''; cf. invariant subgroup), {{math|''H''}}, of a group {{math|''G''}}, is a group remaining [[Invariant (mathematics)|invariant]] under every endomorphism of {{math|''G''}}; that is,
:{{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|commutator subgroup]] of a group is always a fully characteristic subgroup.<ref>
{{cite book
 | title = Group Theory
 | first = W.R. | last = Scott
 | pages = 45–46
 | publisher = Dover
 | year = 1987
 | isbn = 0-486-65377-3
}}</ref><ref>
{{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
}}</ref>

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

=== Verbal subgroup ===
An even stronger constraint is [[Verbal subgroup|verbal subgroup]], which is the image of a fully invariant subgroup of a [[Free group|free group]] under a homomorphism. More generally, any [[Verbal subgroup|verbal subgroup]] is always fully characteristic. For any reduced free group, and, in particular, for any [[Free group|free group]], the converse also holds: every fully characteristic subgroup is verbal.

== Transitivity ==
The property of being characteristic or fully characteristic is [[Transitive relation|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''}}

== 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) × <math>\mathbb{Z} / 2 \mathbb{Z}</math>}}, has a homomorphism taking {{math|(''π'', ''y'')}} to {{math|((1, 2){{sup|''y''}}, 0)}}, which takes the center, <math>1 \times \mathbb{Z} / 2 \mathbb{Z}</math>, 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]]

==Examples==

=== Finite example ===
Consider the group {{math|''G'' {{=}} S{{sub|3}} × <math>\mathbb{Z}_2</math>}} (the group of order 12 that is the direct product of the [[Symmetric group|symmetric group]] of order 6 and a [[Cyclic group|cyclic group]] of order 2). The center of {{math|''G''}} is isomorphic to its second factor <math>\mathbb{Z}_2</math>. Note that the first factor, {{math|S{{sub|3}}}}, contains subgroups isomorphic to <math>\mathbb{Z}_2</math>, for instance {{math|{e, (12)} }}; let <math>f: \mathbb{Z}_2<\rarr \text{S}_3</math> be the morphism mapping <math>\mathbb{Z}_2</math> onto the indicated subgroup. Then the composition of the projection of {{math|''G''}} onto its second factor <math>\mathbb{Z}_2</math>, 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, <math>\mathbb{Z}_2</math>, is not contained in the center, so here the center is not a fully characteristic subgroup of {{math|''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|torsion subgroup]] of an [[Abelian group|abelian group]] is a fully invariant subgroup.

=== Topological groups ===
The [[Identity component|identity component]] of a [[Topological group|topological group]] is always a characteristic subgroup.

==See also==
* [[Characteristically simple group]]

==References==
{{reflist}}

[[Category:Subgroup properties]]

{{Sourceattribution|Characteristic subgroup}}