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:
“When the Devil quotes Scriptures, it’s not, really, to deceive, but simply that the masses are so ignorant of theology that somebody has to teach them the elementary texts before he can seduce them.”
—Paul Goodman (1911–1972)
“Caprice, independence and rebellion, which are opposed to the social order, are essential to the good health of an ethnic group. We shall measure the good health of this group by the number of its delinquents. Nothing is more immobilizing than the spirit of deference.”
—Jean Dubuffet (1901–1985)
“Won’t this whole instinct matter bear revision?
Won’t almost any theory bear revision?
To err is human, not to, animal.”
—Robert Frost (1874–1963)