Abstract algebra is a branch of mathematics that studies algebraic structures through axiomatic definitions. Instead of working with specific numbers or functions, it focuses on the properties and relationships that emerge from fundamental axioms. The main structures include groups, rings, fields, and vector spaces.
A group is one of the most fundamental structures in abstract algebra. It consists of a set G together with a binary operation that satisfies four essential axioms. First, closure means the operation always produces elements within the set. Second, associativity ensures the order of operations doesn't matter. Third, there exists an identity element that leaves other elements unchanged. Fourth, every element has an inverse that combines with it to produce the identity.
A finite group contains a finite number of elements, and this number is called the order of the group. Examples include the cyclic group Z₃ with three elements under addition modulo 3, where the operation table shows how elements combine. Another example is the symmetric group S₃, which has 6 elements representing all possible permutations of three objects, including rotations and reflections.
Infinite groups contain infinitely many elements and can be either countably or uncountably infinite. The integers under addition form a countably infinite group, containing all positive and negative whole numbers. The real numbers under addition form an uncountably infinite group. Another example is the general linear group of invertible matrices under multiplication, which is also uncountably infinite. The key distinction is that finite groups have a specific number of elements, while infinite groups have no bound on their size.
In summary, abstract algebra studies algebraic structures through axioms, with groups being fundamental examples. Groups are classified as finite or infinite based on their number of elements. Finite groups have a specific order and include cyclic groups and symmetric groups, while infinite groups like the integers or real numbers under addition have unbounded elements. Both types have important applications in mathematics, cryptography, geometry, and physics, making group theory essential for understanding symmetry and structure in mathematics.