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Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
Discrete group. The integers with their usual topology are a discrete subgroup of the real numbers. In mathematics, a topological group G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood which only contains that element). Equivalently, the group G is discrete if and only if its ...
The manipulations of the Rubik's Cube form the Rubik's Cube group.. In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
Objects studied in discrete mathematics include integers, graphs, and statements in logic. [1][2][3] By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized ...
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element.
Euclidean group. In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space ; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E (n) or ...
Group action. The cyclic group C3 consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices. In mathematics, many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that ...
In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | is a divisor of | G |, i.e. the order (number of elements) of every subgroup H divides the order of group G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup of a finite ...