The Relevance of Group Theory to Composing Peals of Plain Bob Triples
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Change Ringing is the art of ringing tower or hand bells according to a specific set of rules defined by the British in the 1600s. In particular we examine the group structure in peals of a method called Plain Bob Triples, which are composed of permutations of the hells. This gives us one application of group theory; the symmetric group on n letters or, here, n bells. In this paper we will examine what group structure can be found in a peal of Plain Bob Thiples. We also discuss how peals are composed by factoring the symmetric group on n letters so that the task of testing for repeated permutations is reduced to working in cosets. We show that not only do the lead heads of a peal not have to form a group, they never do. This is a result based almost entirely on the order of the group of lead heads and the fact that the group A 6 is the only subgroup of S 6 with order 360. We also show that the set of leads is the same as the group S 6. Finally we show, as a part of explaining how peals are composed in parts we do a proof to explain why we can be assured of having all the elements of a particular coset if we have just one of them. Finally we discuss two peals whose structure suggests that they are peals composed in parts like most peals, but looks are deceiving, and we find out that they are actually constructed using a very different procedure.