Center (group theory)
In group theory, the center of a group G, denoted Z(G), is the set of elements in G that commute with every other element in G. Formally,
Z(G) = { z ∈ G | zg = gz for all g ∈ G }.
The center is always a subgroup of G. Furthermore, it is always a normal subgroup of G.
Properties:
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Subgroup: The center Z(G) is a subgroup of G. This can be shown by demonstrating closure, identity, and inverse properties.
- Closure: If z1 and z2 are in Z(G), then for any g ∈ G, (z1z2) g = z1(z2g) = z1(gz2) = (z1g) z2 = (gz1) z2 = g (z1z2), so z1z2 is in Z(G).
- Identity: The identity element e of G is always in Z(G), as eg = ge = g for all g ∈ G.
- Inverse: If z ∈ Z(G), then for any g ∈ G, zg = gz. Multiplying by z-1 on both sides gives g = z-1gz. Multiplying by z-1 on the left gives z-1g = z-1z-1gz = z-1gz z-1z = gz-1. Thus, z-1g = gz-1, so z-1 ∈ Z(G).
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Normality: The center Z(G) is a normal subgroup of G. This means that for any g ∈ G and z ∈ Z(G), the conjugate gzg-1 is also in Z(G). This is easily verified: since z is in the center, gzg-1 = g g-1 z = z, and z is in Z(G). A less trivial approach is to show that for any h ∈ G, h(gzg-1) = (gzg-1)h. We know zh = hz, therefore hgzg-1 = gzhg-1 = ghzg-1, which implies Z(G) is normal.
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Abelian Groups: A group G is abelian if and only if Z(G) = G. In other words, if every element commutes with every other element, the center is the entire group.
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Quotient Group: The quotient group G/Z(G) is isomorphic to the group of inner automorphisms of G.
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Trivial Center: A group G is said to have a trivial center if Z(G) = {e}, where e is the identity element.
Significance:
The center provides information about the commutativity within a group. A large center indicates that many elements commute with each other, while a small center (or a trivial center) suggests that the group is far from being abelian. The center is used in various theorems and concepts within group theory, such as the class equation and the study of solvable groups.