Center of a group

Definition

The center of a group, denoted as Z(G), is the set of all elements in a group G that commute with every element of G. This means that for any element z in Z(G) and any element g in G, the equation z * g = g * z holds true. Understanding the center is crucial as it helps identify normal subgroups and plays a significant role in the structure of quotient groups.

5 Must Know Facts For Your Next Test

1. The center Z(G) is always a subgroup of G, as it satisfies the subgroup criteria: it contains the identity element, is closed under the group operation, and includes inverses.
2. If G is an abelian group, then the center Z(G) is equal to G itself since every element commutes with every other element.
3. The center of a group can be used to simplify the study of the group's structure by allowing one to focus on the quotient group G/Z(G).
4. The center can be trivial, consisting only of the identity element, or it can be nontrivial, containing more than just the identity element.
5. The size of the center Z(G) gives insights into how 'non-abelian' the group G is; larger centers indicate that the group has more elements that commute with others.

Review Questions

• How does the center of a group relate to the concept of normal subgroups?
  • The center of a group Z(G) is always a normal subgroup of G because every element in Z(G) commutes with every element in G. This means that for any z in Z(G) and any g in G, the conjugate gzg^{-1} equals z. Therefore, Z(G) retains its structure under conjugation and helps to identify normal subgroups within G, which are crucial for forming quotient groups.
• What is the significance of the center when analyzing non-abelian groups?
  • In non-abelian groups, the center Z(G) provides important information about how 'non-commutative' the group is. If Z(G) consists only of the identity element, it indicates that very few elements commute with others, signifying high non-abelian characteristics. Conversely, a larger center suggests that some elements do commute, revealing a more complex internal structure and affecting how one might construct quotient groups from G.
• Discuss how understanding the center of a group can facilitate understanding quotient groups and their properties.
  • Understanding the center Z(G) allows mathematicians to simplify complex groups into manageable parts through quotient groups. When forming the quotient group G/Z(G), one can effectively reduce the problem of studying G into examining how elements behave modulo those that commute. This process often reveals structural insights and simplifies calculations regarding homomorphisms and other properties of groups. It shows how central elements influence the overall behavior of the group's structure and how these properties can change when considering normal subgroups.

Related terms

Normal Subgroup:

A subgroup N of a group G is called a normal subgroup if it is invariant under conjugation by elements of G, meaning that for every g in G and n in N, the element gng^{-1} is also in N.

Quotient Group: A quotient group G/N is formed by partitioning the group G into cosets of a normal subgroup N, where each coset represents an equivalence class.

Commutative Group: A commutative group, also known as an abelian group, is a group in which the operation is commutative, meaning that for any two elements a and b, a * b = b * a.