Center (group theory)

Cayley table of Dih₄, the dihedral group of order 8.

The center is {0,7}: The row starting with 7 is the transpose of the column starting with 7. The entries 7 are symmetric to the main diagonal. (Only for the identity element is this true in all groups.)

In abstract algebra, the center of a group, $G$ , is the set of elements that commute with every element of $G$ . It is denoted $Z(G)$ , from German Zentrum, meaning center. In set-builder notation,

Z(G) = {z \in G ∣ \forall g \in G, zg = gz}

.

The center is a normal subgroup, $Z(G) ⊲ G$ . As a subgroup, it is always characteristic, but is not necessarily fully characteristic. The quotient group, $G / Z(G)$ , is isomorphic to the inner automorphism group, $Inn(G)$ .

A group $G$ is abelian if and only if $Z(G) = G$ . At the other extreme, a group is said to be centerless if $Z(G)$ is trivial; i.e., consists only of the identity element.

The elements of the center are sometimes called central.

As a subgroup[edit]

The center of G is always a subgroup of $G$ . In particular:

$Z(G)$ contains the identity element of $G$ , because it commutes with every element of $g$ , by definition: $eg = g = ge$ , where $e$ is the identity;
If $x$ and $y$ are in $Z(G)$ , then so is $xy$ , by associativity: $(xy) g = x (yg) = x (gy) = (xg) y = (gx) y = g (xy)$ for each $g \in G$ ; i.e., $Z(G)$ is closed;
If $x$ is in $Z(G)$ , then so is $x -1$ as, for all $g$ in $G$ , $x -1$ commutes with $g$ : $(gx = xg) \Rightarrow (x -1 gxx -1 = x -1 xgx -1) \Rightarrow (x -1 g = gx -1)$ .

Furthermore, the center of $G$ is always a normal subgroup of $G$ . Since all elements of $Z(G)$ commute, it is closed under conjugation.

Conjugacy classes and centralizers[edit]

By definition, the center is the set of elements for which the conjugacy class of each element is the element itself; i.e., $Cl(g) = {g$ }.

The center is also the intersection of all the centralizers of each element of $G$ . As centralizers are subgroups, this again shows that the center is a subgroup.

Conjugation[edit]

Consider the map, $f : G \to Aut(G)$ , from $G$ to the automorphism group of $G$ defined by $f (g) = ϕ g$ , where $ϕ g$ is the automorphism of $G$ defined by

f (g)(h) = ϕ g (h) = ghg -1

.

The function, $f$ is a group homomorphism, and its kernel is precisely the center of $G$ , and its image is called the inner automorphism group of $G$ , denoted $Inn(G)$ . By the first isomorphism theorem we get,

G /Z(G) ≃ Inn(G)

.

The cokernel of this map is the group $Out(G)$ of outer automorphisms, and these form the exact sequence

1 ⟶ Z(G) ⟶ G ⟶ Aut(G) ⟶ Out(G) ⟶ 1

.

Examples[edit]

The center of an abelian group, $G$ , is all of $G$ .
The center of the Heisenberg group, $H$ , is the set of matrices of the form:
${\begin{pmatrix}1&0&z\\0&1&0\\0&0&1\end{pmatrix}}$
The center of a nonabelian simple group is trivial.
The center of the dihedral group, $D n$ , is trivial when $n$ is odd. When $n$ is even, the center consists of the identity element together with the 180° rotation of the polygon.
The center of the quaternion group, $Q 8 = {1, -1, i, -i, j, -j, k, -k}$ , is ${1, -1}$ .
The center of the symmetric group, $S n$ , is trivial for $n \geq 3$ .
The center of the alternating group, $A n$ , is trivial for $n \geq 4$ .
The center of the general linear group over a field $F$ , $GL n (F)$ , is the collection of scalar matrices, ${sIn ∣ s ∈ F \ {0}}$ .
The center of the orthogonal group, $O n (F)$ is ${I n, -I n}$ .
The center of the special orthogonal group, $SO(n)$ is the whole group when $n = 2$ , and otherwise ${I n, -I n}$ when n is even, and trivial when n is odd.
The center of the unitary group, $U(n)$ is $\{e^{i\theta }\cdot I_{n}\mid \theta \in [0,2\pi )\}$ .
The center of the special unitary group, $SU(n)$ is $\left\lbrace e^{i\theta }\cdot I_{n}\mid \theta ={\frac {2k\pi }{n}},k=0,1,....n-1\right\rbrace$ .
The center of the multiplicative group of non-zero quaternions is the multiplicative group of non-zero real numbers.
Using the class equation one can prove that the center of any non-trivial finite p-group is non-trivial.
If the quotient group $G /Z(G)$ is cyclic, $G$ is abelian (and hence $G = Z(G)$ , so $G /Z(G)$ is trivial).

Higher centers[edit]

Quotienting out by the center of a group yields a sequence of groups called the upper central series:

(G 0 = G) ⟶ (G 1 = G 0 /Z(G 0)) ⟶ (G 2 = G 1 /Z(G 1)) ⟶ \dots

The kernel of the map, $G \to G i$ is the $i$ th center of $G$ (second center, third center, etc.), and is denoted $Z i (G)$ . Concretely, the ( $i + 1$ )-st center are the terms that commute with all elements up to an element of the $i$ th center. Following this definition, one can define the 0th center of a group to be the identity subgroup. This can be continued to transfinite ordinals by transfinite induction; the union of all the higher centers is called the hypercenter.^{[note 1]}

The ascending chain of subgroups

1 \leq Z(G) \leq Z 2 (G) \leq \dots

stabilizes at i (equivalently, $Z i (G) = Z i+1 (G)$ ) if and only if $G i$ is centerless.

Examples[edit]

For a centerless group, all higher centers are zero, which is the case $Z 0 (G) = Z 1 (G)$ of stabilization.
By Grün's lemma, the quotient of a perfect group by its center is centerless, hence all higher centers equal the center. This is a case of stabilization at $Z 1 (G) = Z 2 (G)$ .

Notes[edit]

^ This union will include transfinite terms if the UCS does not stabilize at a finite stage.

References[edit]

Fraleigh, John B. (2014). A First Course in Abstract Algebra (7 ed.). Pearson. ISBN 978-1-292-02496-7.

External links[edit]

"Centre of a group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

[1] This union will include transfinite terms if the UCS does not stabilize at a finite stage.

[note 1]

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Monday, August 17, 2020