Center (group theory)

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In abstract algebra, the center of a group G is the set Z(G) of all elements in G which commute with all the elements of G. That is,

Z(G) = \{z \in G \ | \ gz = zg \;\forall\,g \in G\}.

Note that Z(G) is a subgroup of G, because

  1. Z(G) contains e, the identity element of G, because eg = g = ge for all g ∈ G by definition of e, so by definition of Z(G), e ∈ Z(G);
  2. If x and y are in Z(G), then (xy)g = x(yg) = x(gy) = (xg)y = (gx)y = g(xy) for each g ∈ G, and so xy is in Z(G) as well (i.e., Z(G) exhibits closure);
  3. If x is in Z(G), then gx = xg, and multiplying twice, once on the left and once on the right, by x−1, gives x−1g = gx−1 — so x−1 ∈ Z(G).

Moreover, Z(G) is an abelian subgroup of G, a normal subgroup of G, and even a strictly characteristic subgroup of G, but not always fully characteristic.

The center of G is all of G if and only if G is an abelian group. At the other extreme, a group is said to be centerless if Z(G) is trivial, i.e. consists only of the identity element.

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[edit] 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 φg(h) = ghg−1. This 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)\cong \rm{Inn}(G).

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

1 \to Z(G) \to G \to \operatorname{Aut}(G) \to \operatorname{Out}(G) \to 1.

[edit] Examples

[edit] Higher centers

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

G_0 = G \to G_1 = G_0/Z(G_0) \to G_2 = G_1/Z(G_1) \to \cdots

The kernel of the map G \to G_i is the ith center of G (second center, third center, etc.), and is denoted Zi(G). 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.[1]

The ascending chain of subgroups

1 \leq Z(G) \leq Z^2(G) \leq \cdots

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

[edit] 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).

[edit] References

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

[edit] See also