Commutator subgroup

Definition

The commutator subgroup, also known as the derived subgroup, is the subgroup generated by all the commutators of a group. Commutators measure how non-abelian a group is, providing insight into its structure and properties. This subgroup is essential in the study of nilpotent groups, as nilpotent groups have a series of normal subgroups that eventually lead to the trivial subgroup through repeated taking of commutators.

5 Must Know Facts For Your Next Test

1. The commutator subgroup is denoted as [G, G] for a group G and consists of all elements of the form g^{-1}h^{-1}gh for g, h in G.
2. If a group is abelian, its commutator subgroup is trivial, meaning it only contains the identity element.
3. The commutator subgroup is a normal subgroup of the original group and reflects how far a group deviates from being abelian.
4. For nilpotent groups, the commutator subgroup is central, meaning it lies within the center of the group.
5. The process of taking commutators can be repeated to form derived series, leading to insights about solvable groups and their structure.

Review Questions

• How does the commutator subgroup reflect on whether a group is abelian or not?
  • The commutator subgroup provides a direct measure of how non-abelian a group is. If a group is abelian, all elements commute, resulting in its commutator subgroup being trivial—only containing the identity element. This indicates that there are no pairs of elements whose commutation fails, highlighting the absence of any non-abelian structure.
• What role does the commutator subgroup play in identifying nilpotent groups?
  • In nilpotent groups, the commutator subgroup has significant implications as it is always central. This means that it lies within the center of the group, indicating a strong structural characteristic of nilpotent groups. The presence of this centrality reflects their ability to exhibit properties related to solvability and their hierarchical structure through central series.
• Evaluate how the derived series formed from successive commutator subgroups aids in understanding solvable groups.
  • The derived series, formed by repeatedly taking commutator subgroups, plays an essential role in characterizing solvable groups. If this series eventually leads to the trivial subgroup after finitely many steps, it indicates that the original group can be decomposed into simpler abelian factors. This process not only reveals structural insights about solvable groups but also connects them to broader classifications within group theory, emphasizing their importance in understanding group behavior.

Related terms

Abelian Group: A group in which every pair of elements commutes, meaning the operation satisfies the condition that for any two elements a and b, the equation ab = ba holds.

Nilpotent Group: A class of groups where the upper central series terminates in the group itself, characterized by having a non-trivial center and the property that all its normal subgroups are central.

Central Series: A sequence of subgroups of a group where each subgroup is normal in the group, and each quotient is contained in the center of the quotient of the next subgroup.