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Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here.

Dih4 as 2D point group, D4, [4], (*4•), order 4, with a 4 fold rotation and a mirror generator. Dih4 in 3D dihedral group D4, [4,2]+, (422), order 4, with a vertical 4 fold rotation generator order 4, and 2 fold horizontal generator

Groups are very important to describe the symmetry of objects, be they geometrical (like a tetrahedron) or algebraic (like a set of equations). As an example, we consider a glass square of a certain thickness (with a letter "F" written on it, just to make the different positions distinguishable). In order to describe its symmetry, we form the set of all those rigid movements of the square that don't make a visible difference (except the "F"). For instance, if an object turned 90° clockwise still looks the same, the movement is one element of the set, for instance a. We could also flip it around a vertical axis so that its bottom surface becomes its top surface, while the left edge becomes the right edge. Again, after performing this movement, the glass square looks the same, so this is also an element of our set and we call it b. The movement that does nothing is denoted by e.

Given two such movements x and y, it is possible to define the composition x ∘ y as above: first the movement y is performed, followed by the movement x. The result will leave the slab looking like before. The point is that the set of all those movements, with composition as the operation, forms a group. This group is the most concise description of the square's symmetry. Chemists use symmetry groups of this type to describe the symmetry of crystals and molecules.

Let G be the set of bijective mappings of a set S onto itself. Then G forms a group under ordinary composition of mappings. This group is called the symmetric group, and is commonly denoted , ΣS, or The identity element of G is the identity map of S. For two maps f, g in G are bijective, fg is also bijective. Therefore, G is closed. The composition of maps is associative; hence G is a group. S may be either finite or infinite.

If n is some positive integer, we can consider the set of all invertible n by n matrices with real number components, say. This is a group with matrix multiplication as the operation. It is called the general linear group, and denoted GLn(R) or GL(n, R) (where R is the set of real numbers). Geometrically, it contains all combinations of rotations, reflections, dilations and skew transformations of n dimensional Euclidean space that fix a given point (the origin). If we restrict ourselves to matrices with determinant 1, then we get another group, the special linear group, SLn(R) or SL(n, R). Geometrically, this consists of all the elements of GLn(R) that preserve both orientation and volume of the various geometric solids in Euclidean space. If instead we restrict ourselves to orthogonal matrices, then we get the orthogonal group On(R) or O(n, R). Geometrically, this consists of all combinations of rotations and reflections that fix the origin. These are precisely the transformations which preserve lengths and angles. Finally, if we impose both restrictions, then we get the special orthogonal group SOn(R) or SO(n, R), which consists of rotations only. These groups are our first examples of infinite non abelian groups. They are also happen to be Lie groups. In fact, most of the important Lie groups (but not all) can be expressed as matrix groups. If this idea is generalised to matrices with complex numbers as entries, then we get further useful Lie groups, such as the unitary group U(n). We can also consider matrices with quaternions as entries; in this case, there is no well defined notion of a determinant (and thus no good way to define a quaternionic "volume"), but we can still define a group analogous to the orthogonal group, the symplectic group Sp(n). Furthermore, the idea can be treated purely algebraically with matrices over any field, but then the groups are not Lie groups.