What is the importance of group theory

Group theory

In each of these sub-areas, the classification of groups is an important task, and this requires the study of the characters of a group.

The following terms are used in group theory: subgroup, normal divisor, factor group, representation of groups.

In detail: H is a subgroup of the group (G, ·), if H such a subset of G is for the (H, ·) Is a group. The two ever-existing subsets of G are G itself and the one-element group {e}. The term simple group was coined for certain groups that only have “a few” subgroups.

The group classification is possible using the normal divisor and factor group. For the representation of groups Representation of a Lie group.

A group is called finite if it contains finitely many elements. A group that represents a group of rearrangements of a finite set is called a permutation group. The Galois group represents the group of rearrangements of the complex zeros of a polynomial, and its structure provides information about which algebraic equations can be solved in closed form by a solution formula, whereby only the four basic arithmetic operations and the drawing of the n-th roots are permitted as components of the solution formula. The group G is called cyclic if it is generated by a single element, i. H. there is a GG, so that G even the only subgroup H of G is for the GH applies. The group G is called finitely generated if there is a finite subset J of G there so that G even the only subgroup H of G is for the J a subset of H is.

If you have another structure in the set in addition to the group operation G one that must be compatible with the group operation, one arrives at the following terms: A topological group is a group in which a Hausdorff topology is defined in such a way that both multiplication and inverse formation are continuous mappings. If this topology is discrete, the topological group is also called discrete.