Center (group theory)

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,



The center is a normal subgroup, $∀g ∈ G, zg = gz\}$, and also a characteristic subgroup, but is not necessarily fully characteristic. The quotient group, $Z(G) ⊲ G$, is isomorphic to the inner automorphism group, $G / Z(G)$.

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

The elements of the center are central elements.

As a subgroup
The center of G is always a subgroup of $Z(G)$. In particular:
 * 1) $G$ contains the identity element of $Z(G)$, because it commutes with every element of $G$, by definition: $g$, where $eg = g = ge$ is the identity;
 * 2) If $e$ and $x$ are in $y$, then so is $Z(G)$, by associativity: $xy$ for each $(xy)g = x(yg) = x(gy) = (xg)y = (gx)y = g(xy)$; i.e., $g ∈ G$ is closed;
 * 3) If $Z(G)$ is in $x$, then so is $Z(G)$ as, for all $x−1$ in $g$, $G$ commutes with $x−1$: $g$.

Furthermore, the center of $(gx = xg) ⇒ (x−1gxx−1 = x−1xgx−1) ⇒ (x−1g = gx−1)$ is always an abelian and normal subgroup of $G$. Since all elements of $G$ commute, it is closed under conjugation.

A group homomorphism $Z(G)$ might not restrict to a homomorphism between their centers. The image elements $f : G → H$ commute with the image $f (g)$, but they need not commute with all of $f ( G )$ unless $H$  is surjective. Thus the center mapping $$G\to Z(G)$$ is not a functor between categories Grp and Ab, since it does not induce a map of arrows.

Conjugacy classes and centralizers
By definition, an element is central whenever its conjugacy class contains only the element itself; i.e. $f$.

The center is the intersection of all the centralizers of elements of $Cl(g) = {g}$: "$Z(G) = \bigcap_{g\in G} Z_G(g).$"As centralizers are subgroups, this again shows that the center is a subgroup.

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

The function, $G$ is a group homomorphism, and its kernel is precisely the center of $f(g)(h) = ϕg(h) = ghg−1$, and its image is called the inner automorphism group of $f$, denoted $G$. By the first isomorphism theorem we get,

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

Examples
1 & 0 & z\\ 0 & 1 & 0\\  0 & 0 & 1 \end{pmatrix}$$
 * The center of an abelian group, $Inn(G)$, is all of $G/Z(G) ≃ Inn(G)$.
 * The center of the Heisenberg group, $Out(G)$, is the set of matrices of the form: $$ \begin{pmatrix}
 * The center of a nonabelian simple group is trivial.
 * The center of the dihedral group, $1 ⟶ Z(G) ⟶ G ⟶ Aut(G) ⟶ Out(G) ⟶ 1$, is trivial for odd $G$. For even $G$, the center consists of the identity element together with the 180° rotation of the polygon.
 * The center of the quaternion group, $H$, is $Dn$.
 * The center of the symmetric group, $n ≥ 3$, is trivial for $n ≥ 4$.
 * The center of the alternating group, $Q8 = {1, −1, i, −i, j, −j, k, −k}$, is trivial for ${1, −1}$.
 * The center of the general linear group over a field $Sn$, $n ≥ 3$, is the collection of scalar matrices, $An$.
 * The center of the orthogonal group, $n ≥ 4$ is $F$.
 * The center of the special orthogonal group, $GLn(F)$ is the whole group when $\{ sI_{n} ∣ s ∈ F \ {0} \}$, and otherwise $O_{n}(F)$ 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 $\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 ${I_{n}, −I_{n}} |undefined$ is cyclic, $SO(n)$ is abelian (and hence $n = 2$, so $\{I_{n}, −I_{n}\}$ 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:



The kernel of the map $G/Z(G)$ is the $G$th center of $G = Z(G)$ (second center, third center, etc.), denoted $G/Z(G)$. Concretely, the ($(G0 = G) ⟶ (G1 = G0/Z(G0)) ⟶ (G2 = G1/Z(G1)) ⟶ ⋯$)-st center comprises the elements that commute with all elements up to an element of the $G → Gi$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.

The ascending chain of subgroups

stabilizes at i (equivalently, $i$) if and only if $G$ is centerless.

Examples

 * For a centerless group, all higher centers are zero, which is the case $Zi(G)$ 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 $i+1$.