Groups

A group is a set G with a binary operation · on G with the following properties:

Closure

For all elements a, b in G, the result of a · b is an element of G; in other words, performing the operation on two members of the set always produces a member of that set.

Associative property

For all elements a, b, c in G, (a · b) · c = a · (b · c).

Identity

There is a unique element e in G, called the identity of G, such that, for every element in G, a · e = a = e · a.

Inverse

For each element a in G, there is a unique element a⁻¹ in G, referred to as the inverse of a, such that a⁻¹ · a = e = a · a⁻¹.

When dealing with groups, the · operation is sometimes thought of as "multiplication", and so the · is often omitted when writing products. For example, when working with groups, a · e can also be written as ae.

Where the identity element is denoted as 1, exponential notation can be used to indicate multiple applications of the · operation. The usual laws of exponents hold in groups.

While the associative property must hold, the group operation does not have to be commutative; i.e., it does not necessarily have to be the case that ab = ba for all elements a, b of G. Compare with the laws of arithmetic. If the group operation is commutative, however, then the group is called an Abelian group.

An example of a group would be the set of integers together with the operation "+". This can be seen for the following reasons:

• Whenever you add two integers you always get another integer
• Addition is an associative operation, so (a + b) + c = a + (b + c)
• 0 is the identity element; whenever you add 0 to another number, you get the original number
• The inverse of a number is the negative of that number. Whenever you add a number to its negative, you get 0.

There are many other possible types of groups as well. For example, a permutation group is a group consisting of a finite set together with an operation that is a one-to-one function from that set onto itself. To take a simple example, take a set with two elements; say, {1, 2}. There are two permutations of the two elements in this set: (1, 2) → (1, 2) and (1, 2) → (2, 1). If we denote the two permutations by a and b respectively, and denote the permutation operation with ·, then ({a, b}, ·) forms a group with the following operation table:

·	a	b
a	a	b
b	a	b

Informally, we could look at "· a" to mean "leave the set in the same order" (hence a is the identity element for this group) and "· b" to mean "reverse the order of the elements." So, "b · b" means, "take the elements in reverse order and reverse their order." If you reverse the elements twice, you get the elements in their original order, which is what b · b = a means in the table. Similarly, we could define a permutation group for a set of three elements, say {1, 2, 3}; a set of three elements has six permutations.

Of course, when we're permuting elements, it doesn't really matter whether those elements are numbers, letters, or something else. If A is a set containing n elements, then the group of permutations of A is referred to as the symmetric group on n symbols (denoted by (S_n, ·).

Sources used (see bibliography page for titles corresponding to numbers): 46, 51.