It is amazing how different areas of mathematics converge to the same results of what is called today group theory. When we studied group theory in high school, we learned the standard version of the subject. However, the history of development of group theory indicates a considerable amount of independent works in 18th and 19th centuries. According to preeminent mathematicians, the concept group theory evolved from the development of the algebraic equation. Apart from that, they stated that the mathematical group has roots in the number theory and geometry, as well. Therefore, the concept of a group should be considered as an evolving process, a reciprocal movement between historical components, and an event that developed from theoretical methods.

Geometry, over centuries of studies, contributes considerably to the development of the group concept. Euclid’s geometry declared that “two figures are equal if they superposable.” This could be understood in the way that those figures can be displaced, and despite all the changes they may undertake, one can distinguish those which may be regarded as displacements without changing their shape. Hence two figures, which are equal to a third, are equal to each other. From this, it could be implied that the displacement form a “group.”

Other mathematicians like Mobius in 1897 studied geometries from the perspective of the theory that a particular geometry could have invariant properties under a particular group. Steiner’s mathematical work was mainly confined to geometry. In 1832, he laid the foundation of synthetic geometry. He introduces geometrical forms and establishes between their elements a one-to-one correspondence. His work becomes part of the study of transformation groups.

At the end of 18th, Euler studied modular arithmetic. In particular he studied number theory, the law of quadratic reciprocity, which is considered a jewel of arithmetic, a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Although Euler did not study the group theory, he does work in decomposition of an abelian group into subgroups. In mathematics, Euler’s criterion is often used in determining number theory, as a way to express whether a number is a quadratic residue modulo a prime.

Later on Gauss brought Euclid’s work to new levels. The year of 1801 was a fruitful one with comprehensive work on modular arithmetic and abelian groups. During the same year, he presented his quadratic reciprocity law and the prime number theorem.

He studied binary quadratic forms ax2 + 2bxy + cy2 where a, b, c are integers under transformations and substitutions. He partitioned forms into classes then defines a composition on the classes, and later on the associative law emerged.

At the end of 18th Century, the theory of algebraic equations opens the doors to fresh studies of permutations. Lagrange first studied permutations in his paper published in 1770 on the theory of algebraic equations. He studied the cubic and quintic equations and their algebraic approach. In 1799, Ruffini claimed that equations of degree 5 could not be solved algebraically. His work is based on Lagrange’s work, but Ruffini had to introduce the concept of groups of permutations. His proof, however, had errors and in 1802 further evidence was published. He shows that the group of permutations associate with irreducible equations is transitive. The mathematics community, however, did not want to acknowledge his proof that quintics cannot be solved algebraically.

In 1815, Cauchy developed the theory of permutation which was followed by a significant work later on in 1844 which established the theory of permutations as a subject in its own right. For the first time, were introduced the notation of power, positive and negative of permutation and cycle notation and used the term “systeme des substitutions conjuquees” for a group. In 1824, Abel accepted the proof of the insolubility of quintic and Galois in 1831 was the first to work on the algebraic solution of an equation. He related the solution to the structure of a group “le groupe” of permutations related to the equation. In 1832, he discovered the subgroups, which are called normal subgroups today, and he calls the decomposition of a group into subgroups, a “proper decomposition” if the right and left co-set decompositions coincide.

Betti, in 1851, continued the work and published articles relating permutation theory and the theory of equations. In doing so, he proved that Galois’ group associated with an equation was a group of permutations. However, in 1849, an English mathematician called Cayley, published a paper which actually continue the work of Cauchy’s started in 1815. His work was a focus on abstract groups, and realized that matrices and quaternions were groups.

Burnside’s theory of groups of finite order, published in 1897, introduced the 20th century group theory called discrete groups. Continuous groups, developed in the 1900 – 1940 interval with Cartan’s classification of semi simple Lie algebras, Weyle’s theory of representations of compact groups. Finite groups also, gained life of their own, in the 1900 – 1940. Character theory by Frobenius, Burnside, and Schur helped give the work of 19th century to new dimensions.

Contributions to group theory could be seen in 21st century in the work of two mathematicians John Griggs Thompson, who made significant contributions to the inverse Galois problem, and Jacques Tits. The latter, introduced the theory of buildings in algebraic group theory, including finite groups, and groups defined over the p-adic numbers (an extension of the rational numbers achieved by an alternative interpretation of absolute value).

References:

http://en.wikipedia.org/wiki/Quadratic_reciprocity

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Development_group_theory.html

Clifford, Pickover A., The Math Book. From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling 2009