Groups and Subgroups

Any system that satisfies the three conditions above (having an identity, an inverse, and where the operation is associative) is called a ``group'', and the sets of all permutations of various numbers of elements are special groups called ``symmetric groups''. The symmetric group of on $n$ objects is the group consisting of all permutations of $n$ elements, so it contains $n!$ elements--in other words, the symmetric groups get big pretty fast as $n$ gets larger. In this paper we will denote the symmetric group on $n$ elements by $S_n$.

Most practical applications use only a subset of the possible permutations. In Rubik's Cube, for example, although there are 54 little colored faces, it is clear that the ones in the corners will always be in some corner, the ones on the edges remain on the edges, and the ones in the centers of the faces remain centers of faces. Thus in the collection of permutations reachable from a solved cube, there are none that move, say, a corner to an edge.

We will be interested in special subsets of groups that are themselves groups--in other words, a non-empty subset of the permutations so that any product of permutations in the subset is another permutation in the subset.

In our earlier example of $S_n$ (the symmetric group on three elements), there are the following subgroups (including the group that contains only the identity and the entire symmetric group):

$$\begin{eqnarray*} && \{ (1) \},\ \{ (1), (1 2) \},\ \{ (1), (1 3) \},\ \{ (1), (... ... 2) \}, \\ && \{ (1), (1 2), (1 3), (2 3), (1 2 3), (1 3 2) \}. \end{eqnarray*}$$

There aren't any others. If you try to construct some, you'll see what happens. As an example, suppose we try to make one that contains $(1 2)$ and $(1 2 3)$.

It will have to contain $(1 2)^2 = (1)$ and $(1 2 3)^2 = (1 3 2)$. It will also have to contain $(1 2 3)(1 2)= (2 3)$ and $(1 3 2)(1 2) = (1 3)$. But now we've shown that it must contain all the permutations in the symmetric group, so $S_3$ is the group generated by $(1 2)$ and $(1 2 3)$.

If you are a beginner with Rubik's Cube and you want to practice with some operations that jumble the cube but do not jumble it into a nightmare, consider restricting yourself to a subgroup of all the allowable moves. Here are a couple of good examples:

• Only allow moves that consist of $180^\circ$ turns of two opposite faces at the same time. Basically, there are only 3 moves: $R^2L^2$, $U^2D^2$, and $F^2B^2$. These generate some nice patterns as well. This is a very simple subgroup.
• This one is more complicated, but still not too bad. It's basically the same as the one above, except that you're allowed to do single turns of the opposite faces, such as $LR^{-1}$, $UD^{-1}$, and $FB^{-1}$. By repeating these moves you can, of course, get to any position in the subgroup above, but there are many more possibilities.