The Composition of Peals in Parts

Brian D Price

The device which enables us to compose extents with facility is subdivision into parts. This applies from the simplest examples, the plain hunt on 3 bells and Plain Bob Minimus on 4 (both palindromic 3-part extents) to tough problems like composing peals of Stedman Triples. The simplest sets of partheads are cyclic, with a number of working bells rotating among themselves, but there are many more-complex sets of partheads which may be used for composition. In general, we need a group.

Groups were formalised by mathematicians (notably by Lagrange and Cayley) in the 19th century, yet ringers were using them 200 years before that. In ringing we are concerned with groups of permutations, or 'perms', as partheads. The usual term 'rows' will be avoided in this paper, as it will be used in another context. There are in general four criteria for checking a group, but for groups of perms there is but one for which we need to test: closure.

- Closure
- The Fundamental Perm Relation
- Mathematical Terminology
- Method Structure
- Proof
- Perms, Leadheads, Courseheads
- Falseness in Treble-Bob
- Types
- The General Technique
- Marshalling into Types
- Use of a Computer
- The Geometrical Respresentation of Groups
- Groups and "Cosets"
- Enumeration of Groups
- Some Definitions
- The Classification of Groups on up to 7 Working Bells
- Permutation Groups on up to 7 Working Bells
- Logical sequence of Groups on 7 working bells
- Logical sequence of Groups on 6 working bells
- The Group of order 120, transitive on 6 bells
- The Random Generation of Permutation Groups
- Normal Subgroups and Supergroups
- The Outer Automorphism of S6 - a correspondence between groups on 6 or less bells
- A General Way of Generating Permutation Groups
- The Relevance of Normal Subgroups to Composition

- Burnside, W. (1911); Theory of Groups of Finite Order, Cambridge University Press, second edition 1911 (issued 1955 as a Dover reprint).
- Fraleigh J.B. (1982); A First Course in Abstract Algebra, Addison-Wesley, 3rd edition 1982.
- Price, B.D. (1969); Mathematical Groups in Campanology,
*in*The Mathematical Gazette, Vol.LIII No.384, May 1969.

Contributed by: Brian D.Price, 19 Snarsgate Street, LONDON W10 6QP. Contents page for WWW by Don Morrison