Group

A group is a set equipped with an associative binary operation, an identity element, and an inverse for every element. Its axioms capture symmetry and reversible combination.

Binary operation: A rule that combines two elements of a set to produce another element. A group’s operation must combine any two group elements and remain within the set.
Group action: A rule describing how each group element transforms the elements of another set while respecting the group operation. Group actions turn abstract group elements into symmetries of concrete objects.
Symmetry group: A group of transformations that preserve a specified object or structure. The symmetries of an object form a group under composition.
Semigroup: A set with an associative binary operation, without requiring an identity or inverses. It retains associativity but drops the group’s identity and inverse requirements.
Évariste Galois: A French mathematician whose work connected permutations of polynomial roots with solvability by radicals. Galois made groups central to the study of polynomial equations.
Associativity: A property of a binary operation in which regrouping operands does not change the result. Associativity is the group axiom that makes products of several elements unambiguous.
Cyclic group: A group whose elements are generated by repeated application of a single element. A single generator can produce the entire group and reveal its basic structure.
Dihedral group: The group of rotations and reflections that preserve a regular polygon. It gives a concrete finite group combining rotational and reflectional symmetries.
Quasigroup: A set with a binary operation for which each equation obtained by fixing one operand has a unique solution. Quasigroups guarantee division-like solvability without requiring associativity or an identity.
Augustin-Louis Cauchy: A French mathematician who developed foundational results about permutations and finite groups. Cauchy’s work helped establish finite group theory as a mathematical subject.

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