Center
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions
Definition
Symbol-free definition
An element of a group is termed central if the following equivalent conditions hold:
- It commutes with every element of the group
- Its centralizer is the whole group
- It is the only element in its conjugacy class. In other words, under the action of the group on itself by conjugation, it is a fixed point.
- Under the action of the group on itself by conjugation, it fixes everything. In other words, it is in the kernel of the action of the group on itself by conjugation.
The center of a group is the set of its central elements. The center is clearly a subgroup.
Alternatively, the center of a group is defined as the kernel of the homomorphism from the group to its automorphism group, that sends each element to the corresponding inner automorphism. (see group acts as automorphisms by conjugation).
Definition with symbols
Given a group , the center of , denoted , is defined as the set of elements that satisfy the following equivalent conditions:
- for all in
- The conjugacy class of in is the singleton . In other words, under the action of on itself by conjugation, the orbit of is a one-point set -- is a fixed point.
- For the action of on itself by conjugation, acts trivially on everything. In other words, conjugation by fixes every element.
Alternatively, is defined as the kernel of the map given by , where is conjugation by . (see group acts as automorphisms by conjugation).
Group properties
The center of any group must be an abelian group. Conversely every abelian group occurs as the center of some group (in fact, of itself).
Associated constructions
Construction | Name | Description |
---|---|---|
associated quotient-defining function | inner automorphism group | The quotient of a group by its center is isomorphic to the group of inner automorphisms, i.e. the subgroup of the automorphism group comprising those automorphisms that can be expressed using conjugation maps. This is because the map from a group to its automorphism group that sends to is a homomorphism, and its kernel is precisely the center . |
associated ascending series | upper central series | Start with a group . The upper central series of is defined as follows. Consider . Let , in general, be the inverse image in of under the canonical projection . Essentially we are iterating the quotient-defining function that sends a group to the inner automorphism group, and taking the kernel at each step. However, we are pulling back that kernel all the way to . By convention (and commonsense) is the trivial group. A group for which the upper central series terminates in finite length at the whole group is termed a nilpotent group. |
Examples
Below are some examples where the center is a proper and nontrivial subgroup. In other words, these examples exclude abelian groups (where the center is the whole group) and centerless groups (where the center is trivial):
The quotient part in the table below refers to the quotient by the center, which is isomorphic to the inner automorphism group.
Subgroup properties
Properties satisfied
Property | Meaning | Proof of satisfaction |
---|---|---|
central factor | product with centralizer is whole group | |
central subgroup | contained in the center | |
subgroup containing the center | contains the center | |
normal subgroup | invariant under all inner automorphisms | center is normal |
hereditarily normal subgroup | every subgroup is normal in the whole group | center is hereditarily normal |
characteristic subgroup | invariant under all automorphisms | center is characteristic |
quasiautomorphism-invariant subgroup | invariant under all quasiautomorphisms | center is quasiautomorphism-invariant |
strictly characteristic subgroup | invariant under all surjective endomorphisms | center is strictly characteristic |
bound-word subgroup | described by a system of equations | center is bound-word |
purely definable subgroup | can be defined in the first-order theory of the group | center is purely definable |
elementarily characteristic subgroup | no other subgroup that is elementarily equivalently embedded | center is elementarily characteristic |
marginal subgroup | there is a set of words such that the center is precisely the set of elements by which left or right multiplication on any letter of the word does not affect the value (the set of words here is the singleton set comprising the commutator word) | center is marginal |
marginal subgroup of finite type | same definition as for marginal subgroup, but we insist that the set of words be finite (satisfied by the center because we can use a single word). | center is marginal of finite type |
finite direct power-closed characteristic subgroup | in any finite direct power of the whole group, the corresponding direct power of the center is a characteristic subgroup | center is finite direct power-closed characteristic |
direct projection-invariant subgroup | invariant under projections to direct factors | center is direct projection-invariant |
c-closed subgroup | centralizer of some subgroup | center is the centralizer of the whole group |
fixed-point subgroup of a subgroup of the automorphism group | fixed-point subgroup of some subgroup of the automorphism group | via c-closed; explicitly, it is the fixed-point subgroup of the inner automorphism group |
local powering-invariant subgroup | unique root (across the whole group) of an element in the subgroup must be in the subgroup | center is local powering-invariant, also via fixed-point subgroup of a subgroup of the automorphism group |
powering-invariant subgroup | powered for all primes that power the whole group | (via local powering-invariant) |
quotient-powering-invariant subgroup | the quotient group is powered over all primes that the whole group is powered over. | center is quotient-powering-invariant |
Properties not satisfied
The properties below are not always satisfied by the center of a group. They may be satisfied by the center for a large number of groups.
Relation with other subgroup-defining functions
Smaller subgroup-defining functions
- Absolute center: This is the set of elements of the group fixed by every automorphism (not just by every inner automorphism).
- Epicenter: Intersection of inverse images of centers for all central extensions.
- For a group of prime power order, the first omega subgroup (i.e., the subgroup comprising elements of order at most equal to the prime) of the center equals the socle of the whole group, i.e., the join of all the minimal normal subgroups. This subgroup, denoted where is the whole group, is important in many contexts. Further information: socle equals Omega-1 of center in nilpotent p-group
Larger subgroup-defining functions
Subgroup-defining function | Meaning | Proof of containment | Proof of strictness |
---|---|---|---|
second center | inverse image in whole group of center of quotient by center; elements whose induced inner automorphisms commute with all inner automorphisms | ||
Baer norm | intersection of normalizers of all subgroups | Baer norm contains center | center not contains Baer norm |
Wielandt subgroup | intersection of normalizers of all subnormal subgroups | (via Baer norm) | (via Baer norm) |
Related subgroup properties
Property | Definition in terms of center |
---|---|
Central subgroup | contained in the center |
Cocentral subgroup | product with the center is the whole group |
Subgroup containing the center | contains the center |
Effect of operators
Operator | Meaning of application to center | Value |
---|---|---|
fixed-point operator | a group that equals its own center | abelian group |
free operator | a group whose center is trivial | centerless group |
Subgroup-defining function properties
Property name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
reverse monotone subgroup-defining function | Yes | Suppose . Then, . | |
idempotent subgroup-defining function | Yes | For any group , , i.e., the center of the center is the center. |
In groups with additional structure
Topological group
The center of a T0 topological group is always a closed subgroup. Thus, any topologically simple group must be either centerless or abelian.
For full proof, refer: center is closed subgroup
Algebra group
The center of an algebra group must be an algebra subgroup.
For full proof, refer: center of algebra group is algebra subgroup
Computation
The computation problem
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]GAP command
The command for computing this subgroup-defining function in Groups, Algorithms and Programming (GAP) is:Center
View other GAP-computable subgroup-defining functions
To compute the center of a group in GAP, the syntax is:
Center (group);
where group could either be an on-the-spot description of the group or a name alluding to a previously defined group.
We can assign this as a value, to a new name, for instance:
zg := Center (g);
where g is the original group and zg is the center.
References
Textbook references
- Topics in Algebra by I. N. Herstein, ^{More info}, Page 47
- Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 14 (definition introduced in paragraph)
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 50
- Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 52, Point (4.10)
- A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 26, Automorphisms
- Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754, ^{More info}, Page 5 (definition in paragraph, as a special case of centralizer)
- Algebra by Serge Lang, ISBN 038795385X, ^{More info}, Page 14 (definition in paragraph)
- Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 34 (definition introduced in Exercise 11)
- A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, ^{More info}, Page 75, Exercise 52(b) (definition introduced in exercise, as a special case of centralizer, defined implicitly)
- Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, ^{More info}, Page 61
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