A group 
is a finite
or infinite set of group
elements together with a group operation that
satisfies the four fundamental properties of closure,
associativity, the existence of an identity
element, and the existence of inverse elements.
A set is said to be a group "under" this group
operation. Group elements 
, 
,
, ... with group
operation between 
and 
denoted 
form a group if
1. Closure: If 
and 
are group 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 group
element 
.
4. Inverse: There must be an inverse element (a.k.a. reciprocal) of each group element. Therefore, for
each group element 
of 
,
the set contains a group element 
such that 
.
A group is a monoid each of whose group 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 group elements, the group is called a finite group and the number of group elements is called the group order of the group. A nonempty subset of a group that is closed under the group operation and taking inverse elements 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 under
addition or the set of 
invertible
matrices over the real numbers under matrix
multiplication, for a positive integer 
. These last two are examples of Lie
groups.
A nontrivial finite cyclic 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 
modulo 
.
A map between two groups which preserves the identity element and the group operation is called a group homomorphism. If a group homomorphism has an inverse function which is also a group 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 above, 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 group
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 group
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 group representation as one particular incarnation of the group, which may also have other representations. An irreducible representation is a nonzero group representation with no proper nonzero invariant subspaces. For a finite-dimensional unitary representation, this is equivalent to the absence of a simultaneous unitary transformation of its matrices into a block diagonal matrix with two nonempty blocks. For finite groups, the irreducible representations have a number of remarkable properties, as formalized in the group orthogonality theorem.
