Center (group theory)

Cayley table for D₄ showing elements of the center, {e, a²}, commute with all other elements (this can be seen by noticing that all occurrences of a given center element are arranged symmetrically about the center diagonal or by noticing that the row and column starting with a given center element are transposes of each other).
∘	e	b	a	a²	a³	ab	a²b	a³b
e	e	b	a	a²	a³	ab	a²b	a³b
b	b	e	a³b	a²b	ab	a³	a²	a
a	a	ab	a²	a³	e	a²b	a³b	b
a²	a²	a²b	a³	e	a	a³b	b	ab
a³	a³	a³b	e	a	a²	b	ab	a²b
ab	ab	a	b	a³b	a²b	e	a³	a²
a²b	a²b	a²	ab	b	a³b	a	e	a³
a³b	a³b	a³	a²b	ab	b	a²	a	e

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

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 an abelian and normal subgroup of $G$ . Since all elements of $Z(G)$ commute, it is closed under conjugation.

Note that a homomorphism $f : G \to H$ between groups generally does not restrict to a homomorphism between their centers. Although $f (Z (G))$ commutes with $f (G)$ , unless $f$ is surjective $f (Z (G))$ need not commute with all of $H$ and therefore need not be a subset of $Z (H)$ . Put another way, there is no "center" functor between categories Grp and Ab. Even though we can map objects, we cannot map arrows.

Conjugacy classes and centralizers

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

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

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 for odd $n \geq 3$ . For even $n \geq 4$ , 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 $\left\{e^{i\theta }\cdot I_{n}\mid \theta \in [0,2\pi )\right\}$ .
The center of the special unitary group, $\operatorname {SU} (n)$ is ${\textstyle \left\lbrace e^{i\theta }\cdot I_{n}\mid \theta ={\frac {2k\pi }{n}},k=0,1,\dots ,n-1\right\rbrace }$ .
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).
The center of the Rubik's Cube group consists of two elements – the identity (i.e. the solved state) and the superflip. The center of the Pocket Cube group is trivial.
The center of the Megaminx group has order 2, and the center of the Kilominx group is trivial.

Higher centers

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^[1] of $G$ (second center, third center, etc.) and is denoted $Z i (G)$ .^[2] 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

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

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

References

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

External links

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

^ Ellis, Graham (February 1, 1998). "On groups with a finite nilpotent upper central quotient". Archiv der Mathematik. 70 (2): 89–96. doi:10.1007/s000130050169. ISSN 1420-8938.
^ Ellis, Graham (February 1, 1998). "On groups with a finite nilpotent upper central quotient". Archiv der Mathematik. 70 (2): 89–96. doi:10.1007/s000130050169. ISSN 1420-8938.

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

[1] Ellis, Graham (February 1, 1998). "On groups with a finite nilpotent upper central quotient". Archiv der Mathematik. 70 (2): 89–96. doi:10.1007/s000130050169. ISSN 1420-8938.

[2] Ellis, Graham (February 1, 1998). "On groups with a finite nilpotent upper central quotient". Archiv der Mathematik. 70 (2): 89–96. doi:10.1007/s000130050169. ISSN 1420-8938.

[1]

[2]

[note 1]

As a subgroup

Conjugacy classes and centralizers

Conjugation

Examples

Higher centers

Examples

See also

Notes

References

External links