Solvable group

Definition

A solvable group is a type of group in abstract algebra where the group's derived series terminates in the trivial subgroup. This means that if you keep taking commutators of the group, you'll eventually reach a point where you only have the identity element. Solvable groups have nice properties and play an important role in the classification of groups, especially in relation to Galois theory and solving polynomial equations.

5 Must Know Facts For Your Next Test

1. All abelian groups are solvable because their derived series reaches the trivial subgroup immediately.
2. The derived series of a group $$G$$ is defined as $$G^{(0)} = G$$, $$G^{(1)} = [G, G]$$, and continuing with $$G^{(n)} = [G^{(n-1)}, G^{(n-1)}]$$ until reaching the trivial group.
3. Finite groups whose order is a power of a prime are solvable, which is an important result in group theory.
4. The classification of finite simple groups shows that non-abelian simple groups are not solvable, making solvable groups an important part of understanding group structure.
5. Solvable groups are significant in Galois theory as they correspond to certain polynomial equations that can be solved using radicals.

Review Questions

• How does the concept of a derived series help in determining if a group is solvable?
  • The derived series provides a method to examine the structure of a group by repeatedly taking commutator subgroups. If this series eventually reduces to the trivial subgroup, then the original group is classified as solvable. This approach allows for clear identification of solvable groups by examining how far one can go through commutation before reaching only the identity element.
• Discuss how abelian groups relate to solvable groups and what implications this has for their structure.
  • Abelian groups are a special case of solvable groups because their operations are commutative. In an abelian group, all commutators are equal to the identity, leading to an immediate termination of the derived series at the first step. This relationship implies that any properties or results applicable to abelian groups also extend to solvable groups, particularly when analyzing their behavior under various mathematical operations.
• Evaluate the significance of solvable groups within Galois theory and their impact on solving polynomial equations.
  • In Galois theory, solvable groups play a crucial role in determining whether a polynomial can be solved using radicals. When the Galois group associated with a polynomial is solvable, it indicates that the polynomial can be expressed in terms of its roots through radicals. This connection highlights the importance of solvability not just in group theory but also in understanding solutions to equations, further bridging algebraic structures with practical mathematical problems.

Related terms

commutator subgroup: The subgroup generated by all commutators, which are elements of the form $$[a, b] = aba^{-1}b^{-1}$$ for elements $$a$$ and $$b$$ in the group.

simple group: A group that has no normal subgroups other than the trivial subgroup and itself, meaning it cannot be broken down into smaller groups through normal subgroups.

abelian group: A group in which the operation is commutative, meaning that the order of combining two elements does not affect the outcome; specifically, $$a imes b = b imes a$$ for all elements $$a$$ and $$b$$.