A group is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property. The operation with respect to which a group is defined is often called the "group operation," and a set is said to be a group "under" this operation. Elements , , , ... with binary operation between and denoted form a group if
1. Closure: If and are two elements in , then the product is also in .
2. Associativity: The defined multiplication is associative, i.e., for all , .
3. Identity: There is an identity element (a.k.a. 1, , or ) such that for every element .
4. Inverse: There must be an inverse (a.k.a. reciprocal) of each element. Therefore, for each element of , the set contains an element such that .
A group is a monoid each of whose elements is invertible.
A group must contain at least one element, with the unique (up to isomorphism) single-element group known as the trivial group.
The study of groups is known as group theory. If there are a finite number of elements, the group is called a finite group and the number of elements is called the group order of the group. A subset of a group that is closed under the group operation and the inverse operation is called a subgroup. Subgroups are also groups, and many commonly encountered groups are in fact special subgroups of some more general larger group.
A basic example of a finite group is the symmetric group , which is the group of permutations (or "under permutation") of objects. The simplest infinite group is the set of integers under usual addition. For continuous groups, one can consider the real numbers or the set of invertible matrices. These last two are examples of Lie groups.
One very common type of group is the cyclic groups. This group is isomorphic to the group of integers (modulo ), is denoted , , or , and is defined for every integer . It is closed under addition, associative, and has unique inverses. The numbers from 0 to represent its elements, with the identity element represented by 0, and the inverse of is represented by .
A map between two groups which preserves the identity and the group operation is called a homomorphism. If a homomorphism has an inverse which is also a homomorphism, then it is called an isomorphism and the two groups are called isomorphic. Two groups which are isomorphic to each other are considered to be "the same" when viewed as abstract groups. For example, the group of rotations of a square, illustrated below, is the cyclic group .
In general, a group action is when a group acts on a set, permuting its elements, so that the map from the group to the permutation group of the set is a homomorphism. For example, the rotations of a square are a subgroup of the permutations of its corners. One important group action for any group is its action on itself by conjugation. These are just some of the possible group automorphisms. Another important kind of group action is a group representation, where the group acts on a vector space by invertible linear maps. When the field of the vector space is the complex numbers, sometimes a representation is called a CG module.
Group actions, and in particular representations, are very important in applications, not only to group theory, but also to physics and chemistry. Since a group can be thought of as an abstract mathematical object, the same group may arise in different contexts. It is therefore useful to think of a representation of the group as one particular incarnation of the group, which may also have other representations. An irreducible representation of a group is a representation for which there exists no unitary transformation which will transform the representation matrix into block diagonal form. The irreducible representations have a number of remarkable properties, as formalized in the group orthogonality theorem.