Finite group . In modern algebra, the finite group is the one whose cardinal number is n. Precisely, the works of Evaristo Galois began with the study of the finite group of transformations linked to the number of roots of an algebraic equation. During the 20th century, mathematicians have investigated certain aspects of finite groups in great depth, especially local finite group theory, and the theory of solvable groups and nilpotent groups. A complete determination of the structure of all finite groups is too ambitious; the number of possible structures soon becomes overwhelming. However, complete classification of simple finite groups has been achieved. [one]
Finite groups also arise when considering the symmetry of mathematical or physical objects, when those objects admit only a finite number of transformations that preserve the structure. Lie group theory, which can be viewed as a study with “continuous symmetry,” is strongly influenced by associated Weil groups. There are finite groups generated by reflections that act on a Euclidean space of finite dimension. The properties of finite groups can thus play an important role in areas such as theoretical and chemical physics .
Summary
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- 1 Features
- 2 Various cases
- 1 Group of substitutions
- 2 Cyclical groups
- 3 Lie groups
- 3 Linked topics
- 4 References
- 5 Sources
characteristics
- The number of elements in Gis called the order of group G and is represented by | G |.
- If Gis a finite group with an even number of elements (| G | is a pair) then there exists an a in G ≠ e such that 2 = e.
- if Gis a finite group, there exists a positive integer k such that a k = e for any a in G. [2]
Several cases
Substitution group
The symmetric group S N describes all the permutations of N elements. There is N ! possible permutations that give the order of the group. By the Cayley , any finite group can be expressed as a subset of a symmetric group for a given integer N . The alternating group is the corresponding subgroup of the even permutations only.
Cyclical groups
A cyclic group Z N is a group in which all its elements are powers of a certain element a where a N = a 0 = e , the identity element. A typical example of this group is the complex N-roots of unity . Relating to a primitive root of unity , an isomorphism between the two is obtained. This can be done with any finite cyclic group. [3]
Lie groups
Linked Topics
- Algebraic group
- Math operation
- Permutations
