Group Therapy

The study of groups cleanses the soul. This post will define the concept of a group and show a few simple results, after this there will be several different directions to go in for future posts. Familiarity with sets and functions is assumed.

Definition of Group: A group is really a package of two things. It is a set, which we will denote by G, and a function mapping from G\times G to G. For now we shall denote this function by \alpha and call it the multiplication function. The set G and the function \alpha are together required to satisfy the so called Group axioms:

  1. Associativity: For all g,h,k\in G, we have that \alpha(g,\alpha(h,k)) = \alpha(\alpha(g,h),k).
  2. Identity: There exists an element of G called the identity, which we will usually but not always denote by 1_G, with the property that for all g\in G, \alpha(1_G,g) = \alpha(g,1_G) = g.
  3. Inverse: For each element g\in G, there exists another element called the inverse of g, denoted as g^{-1}, which satisfies \alpha(g,g^{-1}) = \alpha(g^{-1},g) = 1_G.

Sometimes one will encounter abelian groups. An abelian group satisfies the additional property that for all g,h\in G, \alpha(g,h) = \alpha(h,g).

The first question to ask is, do any groups actually exist? Our first example of a group will be the infamous trivial group. Let our set G contain only one element. The group axioms force us to choose this sole element to be the identity, i.e. G = \{1_G\}. Since |G| =  1 there is only once choice for \alpha, namely the function that maps (1_G,1_G) to 1_G. With these definitions the trivial group satisfies the group axioms.

A more complicated but still basic example is the unique group with two elements. Let G be the set \{0,1\} and let \alpha be given by \alpha(0,1) = \alpha(1,0) = 1 and \alpha(0,0) = \alpha(1,1) = 0. In this case 0 plays the role of the identity element.

In general groups of all cardinalities exist, and our first basic result about groups is that in any group, there can exist only one element satisfying the properties of the identity. To show this, let (G,\alpha) be a group with identity 1_G and suppose there exists an element e\in G also fulfilling the role of the identity element. By the properties of 1_G we have that e = \alpha(e,1_G). Since e also satisfies these same properties, it must be that \alpha(e,1_G) = 1_G and thus e = 1_G.

In practice, rather than explicitly referring to the function \alpha it is more common to simply write application of \alpha as multiplication. For example, rather than \alpha(g,h) we simply write gh and use parentheses to indicate the order of application. Using this notation we can rewrite the axiom of associativity as requiring that g(hk) = (gh)k. This is a lot neater, though being explicitly aware of the function \alpha is helpful for appreciating the non-triviality of associativity. It will also be useful to refer to the function explicitly when dealing with more than one group, each having their own multiplication functions.

From the definition of a group we can go in several directions. There will be posts delving further into group theory itself, but this post will also serve as a foundation for studying more complex algebraic objects such as rings and fields. Stay tuned.