Definition A binary operation, *, is a rule which associates with each ordered pair of elements (a, b) of a set S, an element a * b of a set T.
For a, b
Example
Sn, define a * b = ab, the product of permutations.
For a, b Z, define a * b = a + b.
For a, b Z, define a * b = ab.
For a, b Z, define a * b = min {a, b}.
For a, b Z, define a * b = a/b.
There is something different about this last example, as a/b is not an integer; i.e. not an element of Z.
We can think of the binary operation as a function which maps ordered pairs (a, b) of S to elements a*b of set T. Some authors insist that T = S, but we allow the more general T here.
Definition A group (G, *) is a set G on which a binary operation is defined which satisfies the following axioms:
G1: (Closure) For all a, b G, a * b G.
G2: (Associative) For all a, b, c G, (a * b)* c = a * (b * c).
G3: (Identity) There exists an e G such that for all a G, a * e = a =
G4: (Inverse) For each element a G there is an element a–1 such that
This completes the list of axioms for a group.
However, a group may satisfy the further axiom:
G5: (Commutative) For all a, b G, a * b = b * a.
A group which satisfies axiom G5 is called an abelian group after the Norwegian mathematician Niels Henrik Abel.
EXTENSION 1
Abel was born in Norway in 1802. As a student he believed that he had found an algebraic solution to the general quintic equation, but soon corrected himself, and published a paper showing that the quintic was not solvable by algebraic means. Much of Abel's research was in analysis. However, he was plagued by poverty for much of his life and tragically died in 1829.
EXTENSION 2
Where are groups found? Some examples:
symmetry
crystallography
quantum mechanics
bell ringing
theory of equations
coding
music
braids, plaits and knots
See Appendix 3 of Budden's book The Fascination of Groups (Cambridge University Press, 1972)