Beyond algebra
The notion of group, due to its structuring nature, is central in algebra. But it is also essential in geometry, where it allows us to understand, classify, relate, study transformations, and even characterize different geometries. Thus, if symmetries don't explain everything, they are often present, sometimes in an invisible manner. But the concept of group does not stop there: it has spread to other domains such as infinitesimal calculus and mathematical physics, where Lie groups have crucial importance.
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Symmetries that leave objects invariant
In geometry, what could be more natural than to seek, for each object, the transformations that leave it unchanged? This means dissecting those transformations and identifying the object's symmetries—and, surprise, groups emerge quite naturally...
Groups of geometric transformations
The notion of a group—abstract and purely algebraic? Not at all! In geometry, it keeps us from being overwhelmed by the apparent profusion and diversity of transformations, and helps us understand the different kinds of symmetry we may encounter.
The Klein group and its many guises
When we first start working with groups, we patiently draw up the tables for those with only a few elements. A one-element group consists solely of the identity element and is therefore unique. Similarly, groups with two or three elements are unambiguously determined. The surprises begin with four elements...
The Erlangen program
On his appointment as a professor at the University of Erlangen in 1872, Felix Klein, then only 23, presented a research program in geometry that has since become known as the "Erlangen Program." The concept of a group lies at its heart.
Lie groups
The groups introduced by mathematician Sophus Lie have become indispensable tools in theoretical physics.
The Cayley diagram
How can we grasp the structure of a finite group at a glance? The Cayley diagram provides the answer!