Group - crowlogic/arb4j GitHub Wiki

A group is a fundamental concept in abstract algebra that captures the notion of symmetry and structure arising from a set with a binary operation. A group is a set $G$, along with an operation (usually denoted as multiplication, but it doesn't have to be the usual multiplication) that combines two elements from the set to produce a third element, also in the set. The operation must satisfy the following four properties:

  1. Closure: For all elements $a$ and $b$ in $G$, the result of the operation, $a * b$, is also in $G$. This property ensures that combining elements in the group always produces an element within the group.

  2. Associativity: For all elements $a, b,$ and $c$ in $G$, the equation $(a * b) * c = a * (b * c)$ holds. This property ensures that the order of performing the operation on multiple elements does not matter, as long as the sequence of the elements is preserved.

  3. Identity: There exists an identity element, denoted as $e$, in $G$ such that for every element $a$ in $G$, the equation $e * a = a * e = a$ holds. The identity element is a neutral element under the operation; when combined with any element, it leaves that element unchanged.

  4. Inverse: For every element $a$ in $G$, there exists an inverse element, denoted as $a^{-1}$, in $G$ such that $a * a^{-1} = a^{-1} * a = e$, where $e$ is the identity element. This property ensures that every element in the group can be "undone" or "cancelled out" by another element under the operation.

Groups are used to describe the algebraic structure of symmetries and transformations in various mathematical and physical contexts. Some examples of groups include:

  • The integers ($\mathbb{Z}$) under addition, with the identity element being $0$ and the inverse of an integer $n$ being $-n$.
  • The set of non-zero real numbers ($\mathbb{R}\setminus{0}$) under multiplication, with the identity element being $1$ and the inverse of a real number $x$ being $\frac{1}{x}$.
  • The set of permutations of a finite set, with the operation being the composition of permutations, the identity element being the identity permutation, and the inverse being the inverse permutation.

Groups provide a powerful framework for understanding many areas of mathematics, such as number theory, geometry, and algebraic topology, as well as physics, where they are used to study symmetries and conservation laws.