📂Abstract Algebra

Center of a Group

Definition¹

The set of elements of a group $G$ that commute with all elements of $G$ is denoted by $Z(G)$ and is called the center of $G$.

$$ Z(G) := \left\{ a \in G : ax = xa \quad \forall x \in G \right\} $$

Explanation

The notation $Z$ comes from the German word Zentrum, which means center. According to the definition, regardless of whether $G$ is abelian or not, $Z(G)$ is an abelian group. Also, trivially, if $G$ is an abelian group then its center is $G$ itself.

$$ Z(G) = G\quad \text{ if } G \text{ is Abelian.} $$

If the center contains only the identity element, it is called trivial. For example, the center of the symmetric group $S_{3}$ is trivial (see below).

Comparison with normal subgroups

Normal subgroup

A subgroup $H$ is called a normal subgroup of $G$ if it satisfies the following property.

$$ gH = Hg \quad \forall g \in G $$

The center of a group resembles a normal subgroup in many ways, and indeed the center is a normal subgroup. The difference is that the center must satisfy the equality with respect to every element, whereas a normal subgroup only needs the equality to hold at the level of sets. In other words, the definition of the center is stricter.

Consider the symmetric group $S_{3}$ as an example. Label its elements as follows.

$$ e = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \end{bmatrix}, \quad \alpha = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{bmatrix}, \quad \alpha^{2} = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{bmatrix} $$ $$ \beta = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{bmatrix}, \quad \alpha\beta = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{bmatrix}, \quad \alpha^{2}\beta = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{bmatrix} $$

Here, the only element that commutes with all elements is the identity, so the center of $S_{3}$ is trivial.

$$ Z(S_{3}) = \left\{ e \right\} $$

On the other hand, the normal subgroup is the alternating group $A_{3}$.

$$ A_{3} = \left\{ e, \alpha, \alpha^{2} \right\} \lhd S_{3} $$

Note the following facts.

• Is the center the smallest normal subgroup? $\to \text{(X)}$
  For an arbitrary abelian group $G$, $Z(G) = G$; in this case $\left\{ e \right\}$ is a normal subgroup, so the center is not the smallest normal subgroup.

• Is the center the smallest nontrivial normal subgroup? $\to \text{(X)}$
  As the example $S_{3}$ above shows, the center can coincide with the trivial normal subgroup.

Properties

• For the symmetric group $S_{n}$, if $n \ge 3$ then $Z(S_{n}) = \left\{ e \right\}$.
• For the alternating group $A_{n}$, if $n \ge 4$ then $Z(A_{n}) = \left\{ e \right\}$.
• For the group of a regular polyhedron $D_{n}$, when $n \ge 3$ holds, if $n$ is even then $Z(D_{n}) = \left\{ R_{0}, R_{180} \right\}$, and if $n$ is odd then $Z(D_{n}) = \left\{ R_{0} \right\}$.

Theorem

(ㄱ) $Z(G)$ is an abelian group.

(ㄴ) $G$ is abelian if and only if $Z(G) = G$.

(a) The center $Z(G)$ of a group $G$ is a subgroup of $G$.

$$ Z(G) \le G $$

(b) $Z(G)$ is a normal subgroup of $G$.

$$ Z(G) \lhd G $$

Proof

(a)

Subgroup test

For a nonempty subset $H$ of a group $G$, if the following two conditions are satisfied then $H$ is a subgroup of $G$.

1. $a$, $b \in H \implies ab \in H$
2. $a \in H \implies a^{-1} \in H$

Step 1: $Z(G) \ne \emptyset$

By the definition of the identity element, for any group $G$ we always have $e \in Z(G)$, hence $Z(G)$ is nonempty.

---

Step 2: $a, b \in Z(G) \implies ab \in Z(G)$

Assume $a, b \in Z(G)$. Then the following holds.

$$ (ab)x = a(bx) = (bx)a = b(xa) = (xa)b = x(ab) $$

Therefore $ab \in Z(G)$.

---

Step 3: $a \in Z(G) \implies a^{-1} \in Z(G)$

Assume $a \in Z(G)$. Then $ax = xa$ holds. Pre- and post-multiplying both sides by $a^{-1}$ gives

$$ \begin{align*} && a^{-1}(ax)a^{-1} &= a^{-1}(xa)a^{-1} \\ \implies && (a^{-1}a)xa^{-1} &= a^{-1}x(aa^{-1}) \\ \implies && xa^{-1} &= a^{-1}x \\ \implies && a^{-1}x &= xa^{-1} \end{align*} $$

Hence $a^{-1} \in Z(G)$.

■

(b)

This follows immediately from the definition of a normal subgroup.

■