Elementary Group Theory
In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation. The notation is normally shortened to the infix notation, or even to .
A group must obey the following rules (or axioms). Let be arbitrary elements of . Then:
- A1, Closure. . This axiom is often omitted because a binary operation is closed by definition.
- A2, Associativity. .
- A3, Identity. There exists an identity (or neutral) element such that . The identity of is unique by Theorem 1.4 below.
- A4, Inverse. For each, there exists an inverse element such that . The inverse of is unique by Theorem 1.5 below.
An abelian group also obeys the additional rule:
- A5, Commutativity. .
Read more about Elementary Group Theory: Notation, Alternative Axioms, Subgroups, Cosets
Famous quotes containing the words elementary, group and/or theory:
“If men as individuals surrender to the call of their elementary instincts, avoiding pain and seeking satisfaction only for their own selves, the result for them all taken together must be a state of insecurity, of fear, and of promiscuous misery.”
—Albert Einstein (1879–1955)
“My routines come out of total unhappiness. My audiences are my group therapy.”
—Joan Rivers (b. 1935)
“In the theory of gender I began from zero. There is no masculine power or privilege I did not covet. But slowly, step by step, decade by decade, I was forced to acknowledge that even a woman of abnormal will cannot escape her hormonal identity.”
—Camille Paglia (b. 1947)