# Group Theory

Group theory is an essential branch of algebra that introduces and explores the concept of groups, playing a pivotal role in the evolution of modern, abstract algebra.

A group is an algebraic framework (G,*) defined by

- a non-empty set G,
- an internal binary operation $$ *:G\times G \rightarrow G $$ that is associative, has an identity element, and guarantees the existence of inverses for all elements.

**Example**. A simple yet illustrative example of a group is the algebraic structure (Z,+), which represents the set of integers under the operation of addition.

Groups form the basis for defining numerous other algebraic constructs, such as rings, fields, vector spaces, and more.

## Origins of Group Theory

After the formulation of solutions by radicals for equations up to the fourth degree, mathematicians pursued similar solutions for equations of higher degrees.

In the early 19th century, it became evident to various mathematicians that equations of the fifth degree could not be solved using radicals.

This era was marked by significant contributions from Ruffini (1813), Abel (1825), and Galois (1832).

The French mathematician **Evariste Galois** associated each algebraic equation with a finite group, demonstrating that an equation is solvable by radicals only if its corresponding group consists of simple Abelian groups.

Galois's work is a cornerstone of group theory and is celebrated as **Galois Theory**.

The modern concept of a group, however, owes much to English mathematician **Arthur Cayley**, who in 1854 was the first to define an "abstract group" by emphasizing the identity element and associative property.

Groups were initially applied in the study of permutations.

Over time, mathematicians recognized the "abstract group" as a versatile concept applicable across various mathematical disciplines (e.g., number sets, differential equations, etc.).

Pioneers in this field during the late 19th century included Kronecker, Lie, and Klein.

By the dawn of the 20th century, group theory had firmly established itself as a groundbreaking local theory within mathematics, complete with its own axioms.

**Note**. The conceptualization of a group as an algebraic structure consisting of a set paired with an associative binary operation, along with an identity element and inverses, is attributed to the influential German mathematician Emmy Noether in the 1920s.

Initial studies focused on finite groups, achieving a significant level of completeness in the field.

The exploration of infinite groups, however, is a more recent development and continues to be an active area of research.

**Note**. Group theory has fostered a modern approach to mathematics that transcends traditional general theories, embracing local theories as well.

And so on.