Binary operation (on a set G)

A function that assigns to each ordered pair of elements of G an element of G

Cayley table, (a.k.a. Operation table)

• A table of results of a binary operation

(written in the form of a multiplication table)

• Every element in the table appears exactly once in each row and column.

(all groups must have this feature!)

Closure

The condition that members of an ordered pair from a set G combine to yield a member of G.

Group

A set G together with a binary operation for which the following properties are satisfied:

(0) Closure

(1)  Associativity: (ab)c = a(bc) for all a, b, c in G.

(2)  Identity: there exists an element e in G for which ae = ea = a for all a in G.

(3)  Inverses: For each element a in G, there is an element b in G for which ab = ba = e.

Closure example

D₄ Cayley Table

(dihedral group 4)