Mathematical Structure -- Groups

A Group has three ingredients --

A set, G, of elements or objects.
A particular element, I, of G, called the identity element.
An operation, called the group operation, which will be denoted by * in this module. The result of applying this operation to two group elements, a and b in that order will be a group element that will be denoted a * b.
The group operation is often denoted by + in which case the result of applying it to the elements a and b is denoted a + b.

The group operation has the following properties, called the group properties --

Associativity -- For any elements a, b, and c in G --

a * (b * c) = (a * b) * c
The identity property -- For any element a in G --

a * I = I * a = a
The inverse property -- for any element a in G there is an element called the inverse of a and denoted

which has the property that

Notice we do not require that for any elements, a and b in G

a * b = b * a

a group that has this additional property is called commutative or Abelian. The term Abelian is capitalized because it is the name of the Norwegian mathematician Niels Henrik Abel (1802 - 1829). A brief biographical sketch of Abel is available at the MacTutor History of Mathematics Archive.

From the short list of group properties we can deduce many other important properties of groups.

Theorem: If x, y, and a are any elements of G and

x * a = y * a

then

x = y

Proof

Missing equation

Notice that if a group is not Abelian then it is not immediately obvious that for any group element a --

Missing equation

The following theorem shows that this is true.

Theorem: If a is any element of G then

Missing equation

Proof

Missing equation

Notice the last line of this proof relies on the first theorem we proved.

Theorem: If x, y, and a are any elements of G and

a * x = a * y

then

x = y

Proof -- This proof is left as an exercise for the reader.

Because the group operation is associative we usually write

a * b * c

instead of either

(a * b) * c

a * (b * c).

Examples:

The set of real numbers with the operation addition is a commutative group.
The set of real numbers with the operation multiplication is NOT a group because 0 has no inverse.
The set of nonzero real numbers with the operation multiplication is an Abelian group.
The set of rigid transformations of the plane with the operation composition is a group. This group is not commutative.
A vector space with vector addition as its operation and the zero vector as the identity element is an Abelian group.

Check Your Understanding

Let Z(n) denote the set {0, 1, 2, ... (n - 1) } -- for example,
Z(6) = {0, 1, 2, 3, 4, 5}
and let a * b be the remainder when a + b is divided by n. Is Z(n) with this operation a group? If it is a group is it commutative? answer
Let F(n) denote the set {1, 2, ... (n - 1) } -- for example,
F(6) = {1, 2, 3, 4, 5}
and let a * b be the remainder when ab is divided by n. Is F(6) with this operation a group? If it is a group is it commutative? Is F(7) with this operation a group? If it is a group is it commutative? For which positive integers n is F(n) a group? answer