symmetrigrupper

Symmetrigruppen are a mathematical concept in group theory that describes a specific type of permutation group. In essence, a symmetrigruppe is a group of permutations that leaves a certain object invariant when acting on it.

The term symmetrigruppe was first introduced by the German mathematician Issai Schur in the early 20th century. Schur was studying the representation theory of finite groups and was particularly interested in the role of permutation groups in this context. He realized that certain groups of permutations exhibited a unique property: when these permutations acted on certain types of objects, such as polynomials or other algebraic objects, they preserved their symmetries.

To be specific, a symmetrigruppe on an object is a group of permutations of that object such that every permutation commutes with every other permutation. In other words, the order in which the permutations are applied does not matter, and the overall effect is the same regardless of the order. This property has useful consequences in various areas of mathematics, particularly in representation theory and algebraic geometry.

The concept of symmetrigruppen has connections to other areas of mathematics, such as Galois theory and the theory of groups of Lie type. Schur's work on symmetrigruppen has had a lasting impact on the development of these areas, and his ideas have influenced generations of mathematicians. While the term symmetrigruppen is less commonly used today, the ideas and concepts have become deeply ingrained in the broader mathematical literature.