Application: Campanology, revisited

Let us put the material on bell-ringing from the previous chapter in the context of group theory. The fact that this can be done so easily is quite remarkable, considering that group theory wasn't developed for at least 100 years later! Generating the plain lead on four bells is analogous algebraically to generating the dihedral group of order $8$, $D_4$. If $a=(1, 2)(3, 4)$, which swaps the first two and last two bells, and if $b=(2, 3)$, which swaps the middle pair, then the plain lead on four bells corresponds to

$$D_4=\{1,a,ab,aba,(ab)^2,a(ab)^2,(ab)^3,(ab)^3a\}.$$

Plain Bob Minimus is equivalent algebraically to generating the symmetric group on 4 elements, $S_4$. Let $a$ and $b$ be as before. If we look at the first column of the Plain Bob Minimus composition, we see that it is nothing more than the dihedral group, $D_4$, which is a subgroup of $S_4$. To generate the second column of $S_4$ we introduce $c=(3,4)$ and let $k=(ab)^3ac$. The second column corresponds to $kD_4$ and the third column to $k^2D_4$. The generation of the Plain Bob Minimus shows that $S_4$ can be expressed as the disjoint union of cosets of the subgroup $D_4$, that is, the cosets of $D_4$ in $S_4$ partition $S_4$. There is an important generalization of this fact, which states:

Theorem 5.14.1   For any group $G$ and any subgroup $H$, the cosets of $H$ in $G$ partition $G$.

As White [Wh] concludes in his paper, he is not suggesting ``that Fabian Stedman was using group theory explicitly, but rather that group theoretical ideas were implicit in (Stedman's) writings and compositions".