Center

Symbol-free definition
An element of a group is termed central if the following equivalent conditions hold:


 * 1) It commutes with every element of the group
 * 2) Its centralizer is the whole group
 * 3) 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.
 * 4) 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 $$G$$, the center of $$G$$, denoted $$Z(G)$$, is defined as the set of elements $$g$$ that satisfy the following equivalent conditions:


 * 1) $$gx = xg$$ for all $$x$$ in $$G$$
 * 2) $$C_G(g) = G$$
 * 3) The conjugacy class of $$g$$ in $$G$$ is the singleton $$\{ g \}$$. In other words, under the action of $$G$$ on itself by conjugation, the orbit of $$g$$ is a one-point set -- $$g$$ is a fixed point.
 * 4) For the action of $$G$$ on itself by conjugation, $$g$$ acts trivially on everything. In other words, conjugation by $$g$$ fixes every element.

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 $$g$$. (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).

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.

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.

Smaller subgroup-defining functions

 * Contains::Absolute center: This is the set of elements of the group fixed by every automorphism (not just by every inner automorphism).
 * Contains::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 $$\Omega_1(Z(G))$$ where $$G$$ is the whole group, is important in many contexts.

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.

Algebra group
The center of an algebra group must be an algebra subgroup.

The computation problem
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.

Textbook references

 * , Page 47
 * , Page 14 (definition introduced in paragraph)
 * , Page 50
 * , Page 52, Point (4.10)
 * , Page 26, Automorphisms
 * , Page 5 (definition in paragraph, as a special case of centralizer)
 * , Page 14 (definition in paragraph)
 * , Page 34 (definition introduced in Exercise 11)
 * , Page 75, Exercise 52(b) (definition introduced in exercise, as a special case of centralizer, defined implicitly)
 * , Page 61