A solvable group is a type of group in abstract algebra where a series of subgroups can be formed such that each is a normal subgroup of the one before, with the quotient groups being abelian. This property is significant in the context of Galois theory, where it helps determine the solvability of polynomial equations by radicals.

Solvable Groups

A solvable group is a type of group in abstract algebra where the group can be broken down into a series of subgroups, each one a normal subgroup of the previous, such that the quotient groups are all abelian. This property is significant because it allows for the application of the theory of solvability to determine whether certain polynomial equations can be solved by radicals.

Solvable Group

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields, focusing on their properties and the relationships between them. It provides a unifying framework for understanding mathematical concepts and solving problems across various areas of mathematics and science.

Abstract Algebra

Series of subgroups

A normal subgroup is a subgroup that is invariant under conjugation by any element of the parent group, meaning it is preserved under the group's internal symmetries. This property ensures that the quotient group, formed by the parent group and the normal subgroup, is well-defined and itself a group.

Normal Subgroup

Quotient groups are a fundamental construction in group theory, formed by partitioning a group into cosets of a normal subgroup, which then form a new group under the operation inherited from the original group. This concept is pivotal in understanding the structure of groups, as it allows for the analysis of groups in terms of simpler, well-understood components.

Quotient Groups

Abelian

Galois theory provides a profound connection between field theory and group theory, allowing the study of polynomial equations through the lens of symmetry. It reveals that the solvability of polynomial equations by radicals is equivalent to the solvability of a specific group of permutations, known as the Galois group, associated with the polynomial's roots.

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