In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in the area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
known as
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a characteristic subgroup is a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
that is mapped to itself by every
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
of the parent
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
. Because every
conjugation map is an
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
, every characteristic subgroup is
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
; though the converse is not guaranteed. Examples of characteristic subgroups include the
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
and the
center of a group.
Definition
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
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
, 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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
; 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
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
, . Then the subgroups, and , 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
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...
(which is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the direct product,
). 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 . Consider and consider the automorphism, ; then is not contained in .
* In the
quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup, , is characteristic, since it is the only subgroup of order 2.
* If is even, the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
of order has 3 subgroups of
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
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 In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
of the parent group, and are therefore not characteristic.
Strictly characteristic subgroup
A ', or a ', which is invariant under
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
s. For
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
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
For an even stronger constraint, a ''fully characteristic subgroup'' (also, ''fully invariant subgroup''; cf. invariant subgroup), , of a group , is a group remaining
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
under every endomorphism of ; that is,
:.
Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups. The
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
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
In mathematics, in the area of abstract algebra known as group theory, a verbal subgroup is a subgroup of a group that is generated by all elements that can be formed by substituting group elements for variables in a given set of words.
For exa ...
, which is the image of a fully invariant subgroup of a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
under a homomorphism. More generally, any
verbal subgroup
In mathematics, in the area of abstract algebra known as group theory, a verbal subgroup is a subgroup of a group that is generated by all elements that can be formed by substituting group elements for variables in a given set of words.
For exa ...
is always fully characteristic. For any
reduced free group
Reduction, reduced, or reduce may refer to:
Science and technology Chemistry
* Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed.
** Organic redox reaction, a redox reacti ...
, and, in particular, for any
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
, 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, , has a homomorphism taking to , which takes the center,
, into a subgroup of , which meets the center only in the identity.
The relationship amongst these subgroup properties can be expressed as:
:
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
⇐
Normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
⇐ Characteristic subgroup ⇐ Strictly characteristic subgroup ⇐
Fully characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphis ...
⇐
Verbal subgroup
In mathematics, in the area of abstract algebra known as group theory, a verbal subgroup is a subgroup of a group that is generated by all elements that can be formed by substituting group elements for variables in a given set of words.
For exa ...
Examples
Finite example
Consider the group (the group of order 12 that is the direct product of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
of order 6 and a
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order 2). The center of is isomorphic to its second factor
. Note that the first factor, , contains subgroups isomorphic to
, for instance ; let
be the morphism mapping
onto the indicated subgroup. Then the composition of the projection of onto its second factor
, followed by , followed by the inclusion of into as its first factor, provides an endomorphism of under which the image of the center,
, 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
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
(or commutator subgroup) of a group is a verbal subgroup. The
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
of an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
is a fully invariant subgroup.
Topological groups
The
identity component
In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element.
In point set topology, the identity compo ...
of a
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
is always a characteristic subgroup.
See also
*
Characteristically simple group In mathematics, in the field of group theory, a group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups. Characteristically simple groups are sometimes also termed elementary groups. Characteristically ...
References
{{reflist
Subgroup properties