Center

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.

Definition with symbols

Given a group $G$, the center of $G$, denoted $Z(G)$, is defined as the set of elements $g$ such that $gx=xg$ for all $x$ in $G$.

Property theory

Reverse monotonicity

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 and iteration

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.

Quotient-idempotence and quotient-iteration

The quotient function corresponding to the center is the inner automorphism group function. This is not idempotent. Quotient-iteration of the center function gives rise to the upper central series.

Subgroup properties satisfied

The center of any group is a bound-word subgroup, and hence, in particular, it is a strictly characteristic subgroup.