Center (group theory)

The center of G is always a subgroup of {{math|G}}. In particular:

1. {{math|Z(G)}} contains the identity element of {{math|G}}, because it commutes with every element of {{math|g}}, by definition: {{math|1=eg = g = ge}}, where {{math|e}} is the identity;
2. If {{math|x}} and {{math|y}} are in {{math|Z(G)}}, then so is {{math|xy}}, by associativity: {{math|1=(xy)g = x(yg) = x(gy) = (xg)y = (gx)y = g(xy)}} for each {{math|g ∈ G}}; i.e., {{math|Z(G)}} is closed;
3. If {{math|x}} is in {{math|Z(G)}}, then so is {{math|x{{sup|−1}}}} as, for all {{math|g}} in {{math|G}}, {{math|x{{sup|−1}}}} commutes with {{math|g}}: {{math|1=(gx = xg) ⇒ (x{{sup|−1}}gxx{{sup|−1}} = x{{sup|−1}}xgx{{sup|−1}}) ⇒ (x{{sup|−1}}g = gx{{sup|−1}})}}.

Furthermore, the center of {{math|G}} is always an abelian and normal subgroup of {{math|G}}. Since all elements of {{math|Z(G)}} commute, it is closed under conjugation.

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

By definition, an element is central whenever its conjugacy class contains only the element itself; i.e. {{math|1=Cl(g) = {g}}}.

The center is the intersection of all the centralizers of elements of {{math|G}}:

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

Consider the map {{math|f : G → Aut(G)}}, from {{math|G}} to the automorphism group of {{math|G}} defined by {{math|1=f(g) = ϕ{{sub|g}}}}, where {{math|ϕ{{sub|g}}}} is the automorphism of {{math|G}} defined by

:{{math|1=f(g)(h) = ϕ{{sub|g}}(h) = ghg{{sup|−1}}}}.

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

:{{math|G/Z(G) ≃ Inn(G)}}.

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

:{{math|1 ⟶ Z(G) ⟶ G ⟶ Aut(G) ⟶ Out(G) ⟶ 1}}.

• The center of an abelian group, {{math|G}}, is all of {{math|G}}.
• The center of the Heisenberg group, {{math|H}}, is the set of matrices of the form: $$\backslash begin\{pmatrix\}$$

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, {{math|D{{sub|n}}}}, is trivial for odd {{math|n ≥ 3}}. For even {{math|n ≥ 4}}, the center consists of the identity element together with the 180° rotation of the polygon.
• The center of the quaternion group, {{math|1=Q{{sub|8}} = {1, −1, i, −i, j, −j, k, −k} }}, is {{math|{1, −1}}}.
• The center of the symmetric group, {{math|S{{sub|n}}}}, is trivial for {{math|n ≥ 3}}.
• The center of the alternating group, {{math|A{{sub|n}}}}, is trivial for {{math|n ≥ 4}}.
• The center of the general linear group over a field {{math|F}}, {{math|GL{{sub|n}}(F)}}, is the collection of scalar matrices, {{math|{{mset| sI_n ∣ s ∈ F \ {0} }}}}.
• The center of the orthogonal group, {{math|O_n(F)}} is {{math|{I_n, −I_n}}}.
• The center of the special orthogonal group, {{math|SO(n)}} is the whole group when {{math|1=n = 2}}, and otherwise {{math|{{mset|I_n, −I_n}}}} when n is even, and trivial when n is odd.
• The center of the unitary group, $U(n)$ is $\backslash left\backslash \{\; e^\{i\backslash theta\}\; \backslash cdot\; I\_n\; \backslash mid\; \backslash theta\; \backslash in\; [0,\; 2\backslash pi)\; \backslash right\backslash \}$.
• The center of the special unitary group, $\backslash operatorname\{SU\}(n)$ is $\backslash left\backslash lbrace\; e^\{i\backslash theta\}\; \backslash cdot\; I\_n\; \backslash mid\; \backslash theta\; =\; \backslash frac\{2k\backslash pi\}\{n\},\; k\; =\; 0,\; 1,\; \backslash dots,\; n-1\; \backslash right\backslash rbrace$.
• 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 {{math|G/Z(G)}} is cyclic, {{math|G}} is abelian (and hence {{math|1=G = Z(G)}}, so {{math|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.

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

:{{math|1=(G{{sub|0}} = G) ⟶ (G{{sub|1}} = G{{sub|0}}/Z(G{{sub|0}})) ⟶ (G{{sub|2}} = G{{sub|1}}/Z(G{{sub|1}})) ⟶ ⋯}}

The kernel of the map {{math|G → G{{sub|i}}}} is the {{math|i}}th center{{Cite journal |last=Ellis |first=Graham |date=1998-02-01 |title=On groups with a finite nilpotent upper central quotient |url=https://doi.org/10.1007/s000130050169 |journal=Archiv der Mathematik |language=en |volume=70 |issue=2 |pages=89–96 |doi=10.1007/s000130050169 |issn=1420-8938|url-access=subscription }} of {{math|G}} (second center, third center, etc.), denoted {{math|Z{{sup|i}}(G)}}.{{Cite journal |last=Ellis |first=Graham |date=1998-02-01 |title=On groups with a finite nilpotent upper central quotient |url=https://doi.org/10.1007/s000130050169 |journal=Archiv der Mathematik |language=en |volume=70 |issue=2 |pages=89–96 |doi=10.1007/s000130050169 |issn=1420-8938|url-access=subscription }} Concretely, the ({{math|i+1}})-st center comprises the elements that commute with all elements up to an element of the {{math|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.This union will include transfinite terms if the UCS does not stabilize at a finite stage.

The ascending chain of subgroups

:{{math|1 ≤ Z(G) ≤ Z{{sup|2}}(G) ≤ ⋯}}

stabilizes at i (equivalently, {{math|1=Z{{sup|i}}(G) = Z{{sup|i+1}}(G)}}) if and only if {{math|G{{sub|i}}}} is centerless.

• For a centerless group, all higher centers are zero, which is the case {{math|1=Z{{sup|0}}(G) = Z{{sup|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 {{math|1=Z{{sup|1}}(G) = Z{{sup|2}}(G)}}.

• Center (algebra)
• Center (ring theory)
• Centralizer and normalizer
• Conjugacy class

{{reflist|group=note}}

• {{cite book

| last1=Fraleigh | first1=John B. | authorlink1=

| year = 2014

| title = A First Course in Abstract Algebra

| edition = 7

| publisher = Pearson

| isbn = 978-1-292-02496-7

}}

• {{springer|title=Centre of a group|id=p/c021250}}

Category:Group theory

Category:Functional subgroups