Definition and examples

A group is a non-empty set G together with an associative binary operation * such that

1.

, ,  e*x = x*e = x

2.

, ,  x*y=y*x=e.

The element e is called an identity of the group. The element y is an inverse of x. However, it may be shown that these are unique elements of the set G and so we say ``e is the identity'' and that ``y is the inverse of x''. Since inverses are unique, then we denote the inverse of x as .

The binary operation we shall often refer to as multiplication (even though it might be function composition) unless we are in a 'familiar' group where the operation is addition. Also, we shall denote multiplication by juxtaposition rather than use a symbol (xy instead of x*y). Further, a group is made up of a set and a binary operation. However, in keeping with standard abuse of notation we shall refer to the group G with the binary operation being understood. Now let's give some examples.

1. The integers under addition form a group. The identity element is 0 and the inverse of n is -n.
2. The rational numbers, real numbers and complex numbers form groups under addition.
3. The positive reals, , under multiplication of real numbers is a group. So too is the set of positive rationals, .
4. The non-zero reals, , under multiplication of real numbers form a group. The identity is 1 and the inverse of x is 1/x. The same is true for the non-zero rational numbers and the non-zero complex numbers. For these two latter groups you might like to find the identity and inverses.
5. The invertible matrices with entries in the real numbers form a group under matrix multiplication.
6. The matrices with entries in the real numbers form a group under matrix addition.
7. A finite set of three elements with the binary operation defined by the following table is a group.

   

   The identity is a, then inverse of b is c and the inverse of c is b. There are 19683 binary operations on a set of three elements. Of these, only three give rise to a group. We'll see later that there is only one such group, structurally.

8. A finite set of four elements with the binary operation defined by the following table is a group.

   

   This is called the Klein-4 group and denoted by . The identity is a. The non-identity elements are self inverse. Structurally, there is only one other group of four elements, defined below.

9. The rotations and reflections of a regular n-gon form a group where the binary operation is function composition. Here the rotations are about an angle around the axis through the center of the n-gon. This ensures that a vertex ends up at the location of a vertex. The reflections are either about lines joining midpoint of an edge to an opposite vertex (when n is odd) or about lines joining opposite vertices or opposite edges if n is even. This group is referred to as the symmetry group of the n-gon (or the dihedral group of order 2n) and is denoted by . Some authors use . Click here for the symmetry group of the square .
10. The integers form a group under addition modulo n. This group is known as .
11. For n a fixed positive integer, n>1, let be the set of positive integers less than n and coprime to n. Under multiplication modulo n this is a group.
12. The eight 2 x 2 matrices with entries in the complex numbers, made up of 4 diagonal matrices (i,-i), (-1,-1), (-i,i),(1,1) and 4 antidiagonal matrices (0,1,-1,0), (0,-1,1,0), (0,i,i,0), (0,-i,-i,0), form a group under matrix multiplication. This group is known as the quaternion group and denoted by .

Multiplcation tables for some specific groups can be accessed from here. These specific examples include other information such as generating sets and epimorphic images.

In these notes, we shall use 1 as the symbol for the identity in a multiplicative group, but 0 for the identity in an additive group. However, be aware that in some examples, both 1 and 0 are elements of the group (such as ) in which case 0 is most likely the identity.

A group G is abelian if for all g and h in G, gh = hg. In addititve notation, this translates to g+h = h+g. The integers, rationals, reals and complex numbers under addition are examples of abelian groups. The symmetry group of the square is not abelian neither is the group of invertible matrices with entries in the real numbers.

• Exponent notation