An introduction to groups

When we studied the Rubik's cube in the previous chapter, recall that one of the criteria was that each move was ``invertible". Another was that the composition of two moves, one after the other, was itself a move. These are, in fact, two of the conditions for the set of all legal moves of a permutation puzzle to form a group. A group is a set $G$ with a binary operation (namely a function $*:G\times G\rightarrow G$) satisfying certain properties to be given later, one of which is that each element has an inverse element associated to it. Before giving the formal definition of a group in §5.3, we will first provide examples of groups that we have seen in earlier chapters. Just as for sets, we must decide on how to describe a group. If $G$ is finite then one way is to list all the elements in $G$ and list (or tabulate) all the values of the function $*$. Another method is to describe $G$ in terms of some properties and then define a binary operation $*$ on $G$. A third method is to give a ``presentation" of $G$. Each of these has its advantages and disadvantages. We shall eventually introduce all three of these approaches. First, we start with some example.