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

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

An Abelian group is a set equipped with an operation that combines any two elements to form a third element, satisfying the properties of closure, associativity, identity, invertibility, and commutativity. Named after mathematician Niels Henrik Abel, these groups are foundational in abstract algebra and are characterized by the fact that the order of operation does not affect the result.

Abelian Groups

Theory of solvability

Polynomial equations are mathematical expressions set equal to zero, consisting of variables raised to whole number powers and coefficients, which provide a foundational structure for understanding algebraic relationships. Solving these equations involves finding the roots or values of the variables that satisfy the equation, often using techniques such as factoring, the quadratic formula, or numerical methods for higher-degree polynomials.

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