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

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.

Definition with symbols

Given a group $G$, the center of $G$, denoted $Z(G)$, is defined as the set of elements $g$ that satisfy the following equivalent conditions:

• $gx=xg$ for all $x$ in $G$
• $C_{G}(g)=G$
• The conjugacy class of $g$ in $G$ is the singleton $\{g\}$

Alternatively, $Z(G)$ is defined as the kernel of the map $G\to \operatorname {Aut} (G)$ given by $g\mapsto c_{g}$, where $c_{g}=x\mapsto gxg^{-1}$ is conjugation by $g$.

Group properties satisfied

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).

Subgroup properties satisfied

The center must satisfy the following subgroup properties:

• Central factor
• Central subgroup
• Normal subgroup
• Characteristic subgroup
• Strictly characteristic subgroup
• Bound-word subgroup
• Hereditarily normal subgroup

The center of a group need not, however, be fully characteristic. For full proof, refer: Center not is fully characteristic

Effect of operators

Fixed-point operator

A group equals its own center if and only if it is an Abelian group.

Free operator

A group whose center is trivial is termed a centerless group.

Subgroup-defining function properties

The center subgroup-defining function is reverse monotone. That is:

Let $H$ ≤ $G$ be groups. Then, $Z(H)$ contains the group $Z(G)$ ∩ $H$.

Idempotence

This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once

The center of the center is the center. This is because the center is an Abelian group, and the center of any Abelian group is itself.

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

Associated constructions

Associated quotient-defining function

The quotient-defining function associated with this subgroup-defining function is: [[Inner automorphism group]]

The quotient of a group by its center is isomorphic to the group of inner automorphisms, This is because the map fro ma group to its automorphism group that sends $g$ to $c_{g}:x\mapsto gxg^{-1}$ is a homomorphism, and its kernel is precisely the center $Z(G)$.

Associated ascending series

The associated ascending series to this subgroup-defining function is: [[Upper central series]]

Start with a group $G$. Consider $Z^{1}(G)=Z(G)$. Let $Z^{i}(G)$, in general, be the inverse image in $G$ of $Z(G/Z^{i-1}(G))$ under the canonical projection $G\to G/Z^{i-1}(G)$. 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 $G$.

By convention (and commonsense) $Z^{0}(G)$ 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.

Computation

The computation problem

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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 aluding to a previously defined group.

We can assign this as a value, to a new name, for instance:

zg = DerivedSubgroup (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
• 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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Definition links

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