Section 6.4
Groups

Monoids in which every element has an inverse deserve a special name.

Definition

A group is a structure (G, *, e, x -> x-1) consisting of a set G, a binary associative multiplication * with unit element e and a unary operation x -> x-1 such that x-1 is an inverse of x.

Note that if G is a group, then (G,*,e) is a monoid. Since every element of a monoid has at most one inverse, we could also have defined a group as a monoid in which every element has an inverse.

In this section we discuss some constructions of groups. Since groups are also monoids, we can consider the same constructions as in Section 6.3. In particular we can consider the (direct) product of two groups because, if (a,b) is an element of the product monoid of two groups, this element has an inverse equal to (a-1,b-1).

Definition 

The product G1 × G2 of the groups G1 and G2 is called the product group. 

Likewise, the product of several groups can be defined. The product of n copies of the same group G is denoted by Gn.

 

As we did for submonoids, we can define

Definition

A subset H of a group G is called a subgroup, if H is a submonoid of the monoid G and the inverse of every element in H is again in H. 

Thus, H is a subgroup of G if the following holds.

  • e  H;
  • a * b  H and a-1  H for all a, b  H.

Just like submonoids are monoids, subgroups are themselves groups: A subgroup contains the unit element, is closed with respect to taking products and contains the inverse of every one of its elements.