Group
a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element.
Groups share a fundamental kinship with the notion of symmetry.For example, a symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other.
The concept of a group arose from the study of polynomial equations
These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry.
The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
Geometry was a second field in which groups were used systematically, especially symmetry groups
The third field contributing to group theory was number theory.
branch of mathematics concerned with the general algebraic structure of various sets (such as real numbers, complex numbers, matrices, and vector spaces), rather than rules and procedures for manipulating their individual elements
Abstract algebra is a part of math which studies algebraic structures. These include:
The manipulations of this Rubik's
Cube form the Rubik's Cube group.
Groups share a fundamental kinship with the notion of symmetry.For example, a symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other.
The concept of a group arose from the study of polynomial equations
A symmetry group
Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, reflections, and translations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetriesThese symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry.
The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
Geometry was a second field in which groups were used systematically, especially symmetry groups
The third field contributing to group theory was number theory.
Abstract algebra
Encyclopedia
abstract algebra branch of mathematics concerned with the general algebraic structure of various sets (such as real numbers, complex numbers, matrices, and vector spaces), rather than rules and procedures for manipulating their individual elements
Abstract algebra is a part of math which studies algebraic structures. These include:
Examples
- Solving equations with many variables. This leads to matrices, determinants and linear algebra
- Finding formulas for polynomial equations. This led to the discovery of groups, as an expression of symmetry.
- Quadratic and higher-degree equations and Diophantine equations - espectially when Fermat's last theorem was proved led to the definition of rings, and ideals.
No comments:
Post a Comment