Solvable group

Definition

A solvable group is a type of group in which the derived series eventually reaches the trivial subgroup. This means that through a series of operations involving commutators, you can simplify the group structure step-by-step until you arrive at the simplest form, which is just the identity element. Solvable groups are significant because they relate to whether certain equations can be solved by radicals, connecting deeply to concepts of field theory and Galois Theory.

5 Must Know Facts For Your Next Test

1. Every abelian group is solvable because the derived series immediately reaches the trivial subgroup since all commutators are trivial.
2. The concept of solvability is crucial in determining whether certain polynomial equations can be solved using radicals; specifically, if the Galois group is solvable, then the equation is solvable by radicals.
3. The derived series of a group G is obtained by iteratively taking commutators: G^{(0)} = G, G^{(1)} = [G,G], G^{(2)} = [G^{(1)},G^{(1)}], and so on.
4. Finite groups with order divisible by a prime number are often solvable; this includes symmetric groups for small n, which helps to understand their structure.
5. Solvable groups play a key role in understanding why the general quintic equation cannot be solved by radicals, as its associated Galois group is not solvable.

Review Questions

• How does the concept of solvable groups relate to the ability to solve polynomial equations by radicals?
  • Solvable groups are directly tied to whether polynomial equations can be solved by radicals. If the Galois group of a polynomial is solvable, it implies that we can construct solutions using roots and radicals. This connection arises from the fact that the structure of these groups allows for a systematic simplification that mirrors how we break down polynomials into simpler factors.
• Discuss the significance of Abelian groups within the context of solvable groups and their properties.
  • Abelian groups are inherently solvable because their commutators are trivial, meaning every element commutes with every other element. This property makes them simple to analyze since their derived series collapses immediately to the trivial group. The study of Abelian groups provides foundational insight into more complex solvable groups and helps establish a clear path for understanding their properties and behaviors.
• Evaluate how solvable groups contribute to our understanding of the unsolvability of the general quintic equation.
  • The unsolvability of the general quintic equation is intricately linked to Galois theory and solvable groups. The Galois group of the general quintic is found to be non-solvable, meaning it cannot be broken down into simpler components through a series of derived steps like solvable groups allow. This property illustrates why certain equations cannot be resolved using radicals and highlights the limitations inherent in algebraic solutions for higher-degree polynomials.

Related terms

Abelian group: A group where the operation is commutative, meaning that the order in which you perform operations does not affect the result.

Radical extension: An extension of a field that can be constructed by adjoining roots of polynomials, often linked to solvable groups.

Galois group: A group associated with a field extension that describes how the roots of polynomials behave under symmetry and transformations.