by Carl Pierer
It is often the case in mathematics that by noticing some kind of symmetry, a problem can be simplified substantially. This makes them very useful. But, much like in art, mathematical symmetries also have an air of beauty and harmony. Perhaps this is one of the reasons why their study is a particularly satisfying branch of mathematics. A symmetry in mathematics is considered as an invariance under certain transformations. Consider, for instance, a square. There are the four rotational symmetries and four symmetries of reflection. While the vertices get permuted under these transformations, the configuration of the triangle is invariant. What this means is that if vertex 1 is joined to the vertices 2 and 4, then no matter what transformation we apply, these vertices will be joined afterwards as well. A transformation that does not keep these joints fixed will not be a symmetry. All these transformations taken together form a mathematical group. These can be, for a lack of a better word, considered as discrete symmetries and belong to the theory of finite groups. A different approach is to consider symmetries of continuous transformations. These are somewhat harder to visualise. Consider a circle. A rotation of the circle by any degree, no matter how small, will preserve the configuration of the circle. This is an example of a continuous transformation. The groups of these kinds of transformation find a natural place in Lie theory.
The person responsible for launching the early investigation of groups of continuous symmetries was Sophus Lie (1842-1899). In the 1860ies, the study of finite groups became a solid part of mathematics; by this time, tools had been developed and mathematicians started using them for various problems. In 1870, Camille Jordan published his Traité des substitutions et des équations algébriques. This was the first detailed study and clarification of Galois Theory. Evariste Galois (1811-1832) had studied algebraic equations and solutions to them. By finding symmetries in the roots of a polynomial and associating a group, he launched a wholly new and fruitful field of mathematical research. His work, possibly due to the highly tumultuous circumstances of his life and his early death, remained in sketches and it was not until others cleared up the ideas that the full impact of the theory was appreciated. Jordan's Traité was an effort to showcase and elaborate on the earlier work on groups by, amongst others Galois and Cauchy. But it also included substantially new contributions, introducing the concept of solvable groups, composition series and proving part of what is today known as the Jordan-Hölder theorem. The work has been credited with being an inspiration for many mathematicians and bringing group theory into the focus of late 19th century and 20th century mathematics.