Solvable groups

Definition

Solvable groups are a class of groups where the derived series eventually reaches the trivial group. This means that there exists a sequence of subgroups starting from the group itself and ending with the trivial group, where each subgroup is a normal subgroup of the previous one and the quotient groups are abelian. Solvable groups are important because they help classify groups based on their structure and properties, particularly in connection with concepts like growth types and amenability.

5 Must Know Facts For Your Next Test

1. Solvable groups are characterized by their ability to be broken down into simpler components through their derived series, which is a key aspect in classifying them.
2. An important example of solvable groups includes all abelian groups, as they trivially satisfy the condition for solvability.
3. The concept of solvability is particularly significant in Galois theory, where it determines whether polynomial equations can be solved using radicals.
4. Not all finite groups are solvable; a classic example of a non-solvable group is the symmetric group on five elements, $S_5$.
5. Solvable groups can exhibit both polynomial and exponential growth depending on their structure and generation properties.

Review Questions

• How do solvable groups relate to the classification of groups based on growth types?
  • Solvable groups can be classified according to their growth types, which depend on how fast the size of their elements grows as we consider larger and larger powers of the group. Some solvable groups exhibit polynomial growth, while others may show exponential growth patterns. Understanding these growth behaviors helps to further categorize solvable groups within broader classifications, allowing for deeper insights into their structure and behavior.
• Discuss how solvable groups contribute to our understanding of amenable groups and their characteristics.
  • Solvable groups play a significant role in understanding amenable groups because every solvable group is amenable. This means that they have a left-invariant mean, which relates to concepts like measure theory and can be connected to various other group properties. Analyzing solvable groups helps researchers uncover necessary conditions for a group to be classified as amenable, thereby linking these two important areas of study in geometric group theory.
• Evaluate the implications of solvability for finite groups and how this influences examples and non-examples of amenable groups.
  • The implications of solvability for finite groups are crucial because while many finite groups are indeed solvable, there exist notable exceptions such as $S_5$, which is non-solvable and hence non-amenable. This distinction affects our understanding of what types of finite groups can be included within classifications related to amenability. By evaluating these characteristics, mathematicians can identify both examples and non-examples that challenge or support existing theories surrounding group classifications.

Related terms

Abelian Groups: Groups in which the operation is commutative, meaning that the order of applying the operation does not affect the outcome.

Nilpotent Groups: Groups that have a central series ending in the trivial subgroup, which implies that all derived subgroups eventually become central.

Group Homomorphism: A structure-preserving map between two groups that respects the group operation.