Set of elements that commute with every element of a group
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 ∈ G | ∀g ∈ G, zg = gz}.
The center is a
normal subgroup, Z(G) ⊲ G, and also a
characteristic subgroup, 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 central elements.
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 ∈ 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) ⇒ (x−1gxx−1 = x−1xgx−1) ⇒ (x−1g = 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.
A group homomorphism f : G → H might not restrict to a homomorphism between their centers. The image elements f (g) commute with the image f ( G ), but they need not commute with all of H unless f is surjective. Thus the center mapping 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. Cl(g) = {g}.
The center is the
intersection of all the
centralizers of elements of 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) = ϕg, where ϕg is the automorphism of G defined by
- f(g)(h) = ϕg(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:
- The center of a
nonabelian
simple group is trivial.
- The center of the
dihedral group, Dn, 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, Sn, is trivial for n ≥ 3.
- The center of the
alternating group, An, is trivial for n ≥ 4.
- The center of the
general linear group over a
field F, GLn(F), is the collection of
scalar matrices, { sIn ∣ s ∈ F \ {0} }.
- The center of the
orthogonal group, On(F) is {In, −In}.
- The center of the
special orthogonal group, SO(n) is the whole group when n = 2, and otherwise {In, −In} when n is even, and trivial when n is odd.
- The center of the
unitary group, is .
- The center of the
special unitary group, is .
- 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 → Gi is the ith center
[1] of G (second center, third center, etc.), denoted Zi(G).
[2] 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.
[note 1]
The
ascending chain of subgroups
- 1 ≤ Z(G) ≤ Z2(G) ≤ ⋯
stabilizes at i (equivalently, Zi(G) = Zi+1(G))
if and only if Gi 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).
See also
Notes
-
^ This union will include transfinite terms if the UCS does not stabilize at a finite stage.
References
External links