Center (group theory)

Conjugacy classes and centralizers

By definition, an element is central whenever its conjugacy class contains only the element itself; i.e. Cl(g) = {g}.

The center is the intersection of all the centralizers of elements of G:

${\displaystyle Z(G)=\bigcap _{g\in G}{Z}_G(g).}$

As centralizers are subgroups, this again shows that the center is a subgroup.

Conjugation

Consider the map f : G → 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: $${\displaystyle \left(\begin{array}{ccc}1& 0& z\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}$$
• The center of a nonabelian simple group is trivial.
• The center of the dihedral group, D_n, is trivial for odd n ≥ 3. For even n ≥ 4, the center consists of the identity element together with the 180° rotation of the polygon.
• The center of the quaternion group, Q₈ = {1, −1, i, −i, j, −j, k, −k}, is {1, −1}.
• The center of the symmetric group, S_n, is trivial for n ≥ 3.
• The center of the alternating group, A_n, is trivial for n ≥ 4.
• The center of the general linear group over a field F, GL_n(F), is the collection of scalar matrices, { sI_n ∣ 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, ${\displaystyle U(n)}$ is ${\displaystyle \left\{e^{i\theta }\cdot I_n\mid \theta \in [0,2\pi )\right\}}$.
• The center of the special unitary group, ${\displaystyle \mathrm{SU}(n)}$ is $\left\{e^{i\theta }\cdot I_n\mid \theta =\frac{2k\pi }{n},k=0,1,\dots ,n-1\right\}$.
• 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:

(G₀ = G) ⟶ (G₁ = G₀/Z(G₀)) ⟶ (G₂ = G₁/Z(G₁)) ⟶ ⋯

The kernel of the map G → G_i is the ith center of G (second center, third center, etc.), denoted Zⁱ(G). Concretely, the (i+1)-st center comprises the elements that commute with all elements up to an element of the ith 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

1 ≤ Z(G) ≤ Z²(G) ≤ ⋯

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

Examples

• For a centerless group, all higher centers are zero, which is the case Z⁰(G) = Z¹(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 Z¹(G) = Z²(G).