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

J.W. Parker, in his article *Mysteries Unveiled* on the composition of Stedman Triples, published in The Ringing World in the early 1950s but dealing with peals composed early this century, clearly knew groups as Regular Course-End Plans and expressed the quality of closure thus:

"...sixty Course-ends are needed, and ... taking any fifty-nine of them, the relationship of the one left out to the fifty-nine must be the same, whichever one ... it may be."We can say more precisely that if A, B and C are three perms in the set, and if D is transposed from C as B is from A, then D must also be in the set, whichever A, B, C are chosen (they need not be all different). Parker's 'sixty' referred to Hudson's group.

A slightly more cryptic reference to the closure of Hudson's group comes in C.D.P. Davies' *Stedman* (1st edition 1903) p.177:

"No matter what variation, reversal, or transposition is attempted... the 60 course-ends of a twin-bob peal.. will always bear precisely the same relation to one another...".

Parker gave his original manuscript of *Mysteries Unveiled* to George Baker of Brighton, who gave it to me just before he emigrated to New Zealand to live with his daughter. I gave it to the Central Council Library.

The term 'transposition' requires defining if we are to use it logically. There are two complementary relations between perms, 'transpositions' and 'transfigures'. Consider the two perms:

A 2 6 4 1 5 3 B 5 3 6 2 1 4

If we regard the relation between them as a transposition, then we are concerned with the bell in 1sts position moving to 4ths, 2nds to 3rds and so on. But if we regard it as a transfigure, then figure 2 is being replaced by figure 5; 6 by 3; etc.

A 2 6 4 1 5 3 C 1 2 3 4 5 6 B 5 3 6 2 1 4 D 5 6 2 1 4 3

If we regard A to B as a transposition, and transpose C (which happens to be rounds) in the same way, we get 562143. But if we were to regard A to B as a transfigure, the answer D would be 254613.

Here we have the fundamental perm relation on which all part composition is built. In the example above, not only are A-to-B and C-to-D the same transposition, but A-to-C and B-to-D are the same transfigure. That this must be so is evident when we consider only the following extracts:

A . . 4 . . . C . . 3 . . . B . . . . . 4 D . . . . . 3

In both transpositions the 3rds place bell has moved to 6ths. The bells which move are 4 in one case, 3 in the other. In both transfigures 4 is replaced by 3.

Put in another way, any transposition and any transfigure (on the same number of bells) are permutable. Starting with perm A, we may apply first the transposition and then the transfigure, or vice versa, in either case ending with perm D.

Returning to our definition of closure of a group of perms, are we to transpose, or transfigure? The answer is, either. If we choose A, B and C at random, and (A to B) defines a transposition which we apply to C, giving D, then we would achieve the same result by swapping perms B and C and applying the transfigure (A to C) to the perm B.

The terminology used in pure mathematics is often at variance with that of the change-ringer. The term "transposition" in mathematics is reserved for a single swap such as 123456 to 123546, whereas in ringing it has been used loosely in either sense of "transposition", "transfigure" of this paper. The meaning of "transposition" used here has the accent on position exchange.

In mathematical textbooks, usually a permutation is given as a transfigure in the form eg (13274)(586), meaning two cycles, one with figure 3 replacing 1, 2 replacing 3 etc, 1 replacing 4; also 8 replacing 5 etc. This is the transfigure (from 12345678) 37218546.

It seems that only through change ringing has the dual nature of transpositions and transfigures been noted (see Price 1969).

At the simplest level, the fundamental perm relation enables the construction of methods, for instance Plain Bob Minimus:

A 1 2 3 4 C 1 3 4 2 1 4 2 3 ------- ------- ------- 2 1 4 3 3 1 2 4 4 1 3 2 2 4 1 3 3 2 1 4 4 3 1 2 4 2 3 1 2 3 4 1 3 4 2 1 4 3 2 1 2 4 3 1 3 2 4 1 B 3 4 1 2 D ? ? ? ?

The changes between consecutive perms are (because of the rules of change ringing) transpositions, and in the first (left-hand) lead the movement from A to B is a series of five changes which together amount to a transposition moving the 1st bell to 3rds, 2nd to 4ths etc. C is the second lead-head and we must regard it as derived from A by a transfigure, figure 2 being replaced by figure 3 etc.

What will perm D be? We can either regard it as a transposition from C, or as a transfigure from B. Thus, moving downwards, the leads have a transposition structure, while across the rows is a transfigure structure of the cyclic interchange of figures 2, 3, 4. All methods and principles conform to this pattern.

The above example illustrates, in an elementary way, an important application of the fundamental perm relation.

A 1 2 3 4 B 1 3 4 2 C 1 4 2 3 ------- ------- ------- D 2 1 4 3 E ? ? ? ? F

We have started with a simple group of perms (A, B, C) as lead-heads. They form the cyclic group of order 3 on 234, with the treble as fixed bell. From A, we move downwards with the transposition that crosses two pairs of bells, to give a perm D which is new. Will perm E, when we have constructed it by the fundamental perm relation, be different from any of A, B, C, D we already have?

We can prove that it must be different. E cannot be the same as D, because D-to-E is the same transfigure as A-to-B and that pair are different. Suppose that E is the same perm as one of A, B, C. Then it belongs to their group. By the closure property, as A, B and E belong the the group, so must D. But D is a new perm, not one of A, B, C. We have reached a contradiction, and so the supposition is false. E too must be a new perm.

This logical argument continues, until the extent of 24 perms is achieved. Perms in the 2nd and 3rd columns will fall into place once a perm is found for the first column which is not already in the rows above it. G will be chosen as a new perm under D, and so on.

The composer does not usually operate the above process with the individual perms of a peal, although on occasions this happens, for instance in creating triples principles similar to John Carter's Scientific Triples. The extent of 5,040 perms is marshalled by a certain group of order 168 (i.e. it has 168 perms as partheads), and a principle is found with 30 perms to the section, such that the 168 sections comprise the true extent. As the group has some transfigures which are 7-part cyclic, principles can be found which have a 7-part plain course of 7x30=210 perms. The trouble starts when one attempts to find acceptable calls for linking the 24 courses together.

Usually, the composer thinks in terms of leadheads, or courseheads (exceptionally, Stedman has sixends). For instance, with Grandsire Triples one might deal with leadheads. In one way of composing Grandsire (the easiest way) one deals only with the 360 in-course leadheads, being half the extent on the six working bells 234567, and 360x14=5,040. In order to compose Grandsire Triples in parts, a group of leadheads is chosen from the 360, which will then have the effect of marshalling the 360 leads in parts.

With many methods, such as triples and plain even-bell methods, the leads are not false internally so that the leadheads may be used (with certain reservations) without involving truth, but treble-bob methods are liable to internal falseness. However, the same principles still apply, the extra matter of falseness of leads obeying the same general structure. Consider the general case of composition of a treble-bob major method with no restrictions on the tenors. There are 5,040 different leadheads, and we have proposed a group of them as partheads:

Pa Pb Pc Pd Pe Pf Pg ... Pz -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- Aa Ab Ac Ad Ae Af Ag ... Az -- -- -- -- -- -- -- -- -- xx -- -- yy -- -- -- Ba Bb Bc Bd Be Bf Bg ... Bz -- -- -- xx -- -- -- zz -- -- -- -- -- -- -- -- .. .. .. .. .. .. .. ... ..

The diagram indicates in general terms an array of leads of the method. Pa, Pb, Pc... Pz are the group of partheads. Aa is a leadhead not in the top row, and Ab, Ac, ... the other leads initiated by Aa; and so on. Suppose that xx is a perm which occurs in two different leads headed by Ab, Bd. yy is a perm in a position corresponding to the first xx. Consider the chain of transpositions (downwards) and transfigures (across):

Pb --- Pe Ab Ae xx yy=zz Bd Bz Pd -------Pz

The left-hand column of transpositions is made up of Pb-Ab (across leadheads), Ab-xx (part of lead Ab), xx-Bd (part of lead Bd reversed), Bd-Pd (across leadheads). The three leadheads Pb, Pd, Pe are in the group of partheads, which is closed, hence we construct the fourth parthead Pz by the fundamental perm relation, and apply the transposition of the first column to Pz in reverse, arriving at perm zz. The loop closes with perm zz, which must be identical to perm yy, because xx-yy and xx-zz are the same transfigure.

We have proved that if a lead in one row of the array is false with a lead in another, then every lead in the first row is false with a lead in the other. It may be that this applies to two leads in the same row, in which case every lead in the row will be false with another in the same row, and none of the leads in the row can be used for a part composition (though a selection might be spliced in later).

A similar argument applies for calls. If for instance a lead in one row gives by a bob a lead in another, then every lead in the first row when bobbed gives a lead in the other. These relations make composition in parts possible, and the use of a group to initiate the process is a sufficient condition. It is also in practical terms a necessary condition.

Since every lead in a row of the array of leads has the same properties as any other, in respect of both calls and falseness, we can think of the whole row as an entity for the purposes of composition, and cease to consider individual leads. In the case of treble-bob major, if for instance we are using the group of 7 cyclic partheads, then the 5,040 leads will be marshalled by the group into 720 rows, which we term 'types'. We will be able to say, for instance, that type 361 gives type 240 by a bob, or that type 31 is false with type 622. These numbers stand for rows of leads, and the task of the composer (perhaps delegated to a computer) is to give arbitrary type numbers to the rows of perms, which henceforward will be used for plotting calls and falsity.

That composers such as W.H. Thompson, J.W. Parker, Law James and Pitman were familiar with these concepts is quite clear. Apart from the fact that they could not have achieved what they did without the technique, typing is explicit in Parker's "Mysteries Unveiled" for Stedman sixes using Hudson's group of 60 courseheads. It is also to be found in the later chapters of Snowdon's Stedman (1903 edition), in connection with the composition of twin-bob peals in parts.

Thus the general technique of part composition is:

- Decide on a group for the partheads. It must be a sub-group of the extent of the sections (leads, courses or sixes) of the method being composed (i.e. all partheads belong to this extent) and its order will be a factor of the order of that extent.
- Use the group to marshall the sections into their row sets.
- Give each row set an arbitrary type number.
- Make a table stating how calls (including Plain) transform one type into another.
- If internal falseness is involved, make a table giving the list of types which are false with each type.
- A peal is produced by trial and error, attempting to string types into a continuous loop using the call table for systematically choosing the junctions between types, and checking each new type, that it does not repeat with the types already in the string, and if necessary is not false with them by use of the false table.
- A loop solution, if lucky, will be a peal without further ado. If very unlucky it will issue into rounds in one part, with the extent in a number of separate touches. Often, several touches each in parts will result, which have to be joined by special calls.

The process (b) above may be achieved in a number of ways. Firstly, it is possible to write out all the sections in sets, which is what A.J. Pitman reputedly did in his signal box at Port Talbot. It is evident from the writings of J.W. Parker that he also did this, as no doubt many other composers did. We can program a computer to do the job. W.H. Thompson and others used geometrical methods to display types, and their Q-sets.

In certain special cases the process of finding types is simplified by fixed bells and composition is more rational a process. This occurs if the group used is the full extent, or the extent of the in-course perms, on a number of working bells, leaving the rest fixed. For instance, Bristol Surprise Major with the tenors unparted has 7 working bells including the tenors, but only 5 may be affected by calls. If the 60 in-course perms of these five be used as the group, then there will be 40,320/60 = 672 different types of perms in the extent, and these will correspond to the 336 permutations of bells 1, 7 and 8 among the eight positions, each permutation defining one in-course and one out-of-course type. But it so happens that a plain course of Bristol has no repetition of these types (unlike most Surprise methods) hence the 60 courses are "clean proof". As a course contains 224 of the 672 possible types of rows, or one-third, that will be the proportion of the total extent attainable with fixed tenors. This happens likewise with a bobbed course of Cambridge.

The same argument applies to the composition of any plain triples method, where the fixed treble combined with in- and out-of-course gives 14 types of perms, a lead contains one of each type, and the extent is obtained simply by the 360 in-course lead heads.

Such an often-used process is a special case of the use of groups, which in general is not so easily comprehended.

These processes may now be carried out by computer. How this may be done is outside the scope of this paper; the starting point is the writing of a subroutine to identify a perm in an extent as a serial number, and another to carry out the reverse process.

The Geometrical Representation of Groups

Groups are fascinating in that the same group can occur in different guises; a group of perms can appear as a set of rotations of a geometrical shape. The simplest case is that of the cyclic group, where (say) 5 figures are rotating: 12345, 23451, 34512 etc and the diagram is a pentagon in two dimensions, with the figures 1 to 5 written on the five corners. Rotation of the pentagon corresponds to the transpositions (or transfigures) of the group of perms. A not-quite-so-simple case is that of the pentagonal prism, the five parallel edges labelled 1 to 5, where one may turn the solid over by a rotation of half a turn about an edge-side axis and then rotate it again in fifths of a turn; the result is the 'dihedral' group of perms 12345, 23451, 34512 etc together with 54321, 43215, etc, ten perms in all, a group which also contains 'two pairs swapping' transfigures.

The author considers only real transformations on geometrical solids or plane figures, i.e. transformations which can be carried out by rotating a solid or a plane figure so that the actual rows may be written on the figure and the transpositions studied by labelling the vertices, edges etc. with the bells. Books on mathematical groups sometimes consider reflections, which cannot be performed practically. Thus Fraleigh (p.98) quotes the group S4 [4.01] as being the group of symmetries of the regular tetrahedron, but he is including reflections and other transformations which cannot actually be carried out on a solid tetrahedron. The group of rotations, all of which can be carried out, form the alternating group A4 [4.02] and in this case the vertices may be labelled 1 to 4 and the twelve +ve rows on four bells actually written on the model in the face-corners, so that rotations transpose the figures in the rows as they do on the numbered vertices. Burnside (Chapters XVIII, XIX) similarly emphasises "real rotations" in his treatment of the same topic.

Similarly, Fraleigh gives the symmetry group of the square as the dihedral group D4 [4.03], and so it is if reflections are considered. I prefer to think of the rotation group of the square as the cyclic group C4 [4.05] because if one draws a square and keeps it flat, the bells 1 to 4 can be placed at the vertices and the four rows can be written in the four corners so that rotations of 90, 180 or 270 degrees correspond to transpositions. If one wants the dihedral group D4 [4.03] then the square is made solid into a square prism (but not a cube) with short new edges, and the bells 1 to 4 written on these edges so that the extra four rows are on the reverse square face. It is now possible to make various rotations of 180 degrees to turn the prism over, effecting 2-part transpositions of the bells. This group is more complex than at first appears - there are three different kinds of 2-part transpositions in it, a situation which makes complications when normal subgroups are considered.

Exceptionally, there are certain groups which can be put on an infinite plane lattice e.g. [5.03], [6.12], [6.29]; a portion may be drawn on paper and labelled with the rows and bells, and the rotations or translations may be done practically. [6.29] turns out to be isomorphic with [6.12] (see section on the outer automorphism of S6) and will have a similar graphical structure. The infinite lattice may be made finite by cutting out a representative portion and joining up the corresponding edges, but the result is a torus (a ring with a hole) which is not practicable to handle, and impossible to rotate (topologically speaking) as a rigid figure. See Burnside Chapter XVIII for a full treatment of graphical representation.

W.H. Thompson used the dodecahedron (a solid with 12 pentagonal faces) to plot Grandsire Triples, but the record of how he did it seems to have been lost. The dodecahedron has 30 edges, which fall into sets of 6, the edges in each set being either parallel or perpendicular. If the 5 sets be labelled 1 to 5 (6 edges being labelled with each number) then the possible rotations of the solid transpose the figures in such a way that all 60 in-course permutations are produced (group 5.02). They can be written symmetrically on the 60 face-corners. The solid can be used for composing touches of minor with bobs only (in-course permutations of 23456) and singles can be brought in by an overlaid pattern. Grandsire Triples has 6 working bells, so Thompson may have used the device of a fixed 7th bell in order to plot perms of 23456 on the solid.

However, the dodecahedron has 12 faces, in 6 opposite pairs, and if the 6 pairs are labelled 1 to 6, a group of order 60 on 6 working bells is produced, which is Hudson's group, the basis for the usual twin-bob peals of Stedman Triples. This group (6.05) is apparently different from the group of 60 in-course perms on 5 bells, and so it is to the ringer, but to the mathematician the two are manifestations of the same abstract group (see section on the Outer Automorphism of S6, where groups 5.02 and 6.05 are duals). It is possible that Thompson used the Hudson group for displaying Grandsire.

Thompson's diagrams in the appendix of the first edition of Grandsire are topological diagrams of the lead types of Grandsire Triples, showing the Q-set linkages between them. The different diagrams are for different generating groups. Thompson's groups were all positive ones - he did not explore the more difficult use of mixed groups (as in the 12-part peal, and a 4-part peal, by J.J. Parker).

In the general method of composition, the extent (of rows, leadheads, partheads etc) forms the overall group, and the group we choose for composition is a subgroup of this overall group. This subgroup is then transfigured to give the type structure. In general mathematical terms, the types are cosets. Thus in the example of Plain Bob Minimus on p.2, the overall group is the extent of 24 rows, the group chosen is of order 3 (the three positive perms with the treble leading, including rounds) and the succeeding rows (which do not contain rounds) are cosets, transposes of the chosen group.

Mathematically speaking, a group is a set of operations, and the groups used in change ringing are actually a set of transfigures existing between the perms of a set, and not the perms themselves. Here a confusion arises. To write down a transfigure we need to write a perm such as 421365. We are implying that rounds, 123456, is the "standard perm" and that the transfigure 421365 is a set of figure changes from 123456 to 421365. But we do not necessarily have to have rounds itself as a starting point.

If we take the second row of Minimus on p.2:

2143 3124 4132

then the internal transfigures between these three (including that of the perms with themselves) form the same group of order 3 as for any other row. Rounds itself does not have to be present in a group of perms for ringing purposes (though usually it is). For instance, Hudson's sixty courseheads in Stedman Triples conventionally contain the course 2314567 (the quick sixend of the usual start of Stedman) but no course 1234567.

In order to write down a transposition or a transfigure, we need an arbitrary perm as reference, and this is the role of rounds.

Composers are interested in what groups of perms are available for composition. Following is an attempt to give a list of all possible groups on up to 7 working bells. 7 is as far as we need to go, as on the higher numbers of bells it is customary to keep many of them fixed in peals. And above triples, it is not essential to ring the extent in a peal; hence the use of a group would not appear to be necessary, but nevertheless in practical terms it is.

Groups 'exist', one discovers them but one cannot make them to order. One cannot make a Scientific Triples-type principle by taking a group of order 315 and marshalling 16 types of perms, because a group of order 315 on 7 working bells does not exist.

The list was compiled by a combination of various approaches:

- The geometrical forms of groups help, for they provide a way of 'lateral thinking' which can give rise to isomorphic permutation groups by labelling faces, edges, vertices with numbers.
- A technique of random generation of permutation groups was devised, as explained later in this paper.
- A correspondence between groups on 6 or less bells (the outer automorphism of S6) provided a useful check.

(those preceded by an asterisk are probably confined to this paper)

The **Alternating Group** on n bells, of order always half of the extent, consists of all the positive (in-course) perms. Thus the alternating group on 5 bells, "A5", consists of the 60 positive perms.

The **Degree** of a permutation group is the number of bells in it.

The **Dihedral Group** on n bells has order 2n, and consists of the cyclic group of n perms formed from rounds by rotating the bells, together with the reverse of these perms. Thus on 5 bells the Dihedral group is:

12345 23451 34512 45123 51234 15432 21543 32154 43215 54321

Geometrically it corresponds to a prism having its end faces regular n-gons, its set of n parallel edges being labelled 1 to n. The Dihedral group on 3 bells, of order 6, is the same as the extent S3.

The **Extent **(or **Symmetrical Group**) on n bells consists of n! perms. Thus on 6 bells the extent, "S6", has order n! = 6x5x4x3x2x1 = 720.

The ***Involute** of a cycle; a cyclic group can be given a geometrical form by its n figures being spaced around a circle. If, starting from any figure, another cycle is created by stepping round every mth figure (m having no common factor with n) an involute cycle is created.

The **"Pairs of Pairs" Group** [4.04] is a group of order 4 on 4 bells, consisting of rounds plus the three perms produced from rounds by swapping pairs of pairs:

1234 2143 3412 4321

It has special properties in relation to the production of palindromic compositions, and is one of the forms of Klein's "Vieregruppe" (4-group). It is a normal subgroup of the alternating group [4.02] and is the only case of a normal subgroup of an alternating group on any number of figures (Burnside p.181). There are four isomorphs of the group in the list: [4.04], [4.06], [6.27], [6.35]. They all have the same geometrical representation, the group of rotations of a cuboid. Of these, only the "pairs of pairs" permutation group is transitive.

A **Palindrome** in campanology is a sequence of transpositions constituting the plain course of a method or principle, or a touch or peal of such, which is unaltered when the order of the transpositions is reversed. (A literary palindrome is a sentence which spells backwards the same as forwards, eg "A man, a plan, a canal, Panama").

***Parent Group.** Parent Group is the inverse concept of Primary Subgroup, i.e. it is a group of higher order containing the group in question as a subgroup, there being no intermediate subgroup.

**Primitive Group**. When the bells of a transitive permutation group can be divided into sets, the bells of which are either interchanged among themselves or changed into the bells of another set, by every permutation of the group, the group is termed Imprimitive; if this division is not possible, the group is Primitive.

**Subgroup.** A Subgroup of a group is a set of perms which are contained in the group, which themselves form a group. The order of the subgroup must be a factor of the order of the group.

***Primary Subgroup**. If B is a subgroup of A, and C is a subgroup of B, then C must be a subgroup of A (i.e. Subgroup is a transitive property). C is a primary subgroup of A if no intermediate subgroup, such as B in the example, exists. In the group listing the primary subgroups are given. All the subgroups of a group may be found by looking up primary subgroups of the primary subgroups, and so on. Further, the primary subgroups do themselves have at least one subgroup - if not, the term is just plain *subgroup*.

**Normal Subgroup**. This is the more recent term in group theory for "self-conjugate subgroup" (Burnside p.30). Since normality is not a transitive property, it is not possible to abbreviate the lists of normal subgroups as with Primary Subgroups and Parent Groups for subgroups; so that the lists of normal subgroups must be given in full. An explanation of the relevance of normal subgroups to change ringing, with a "ringer's definition", is the last topic of this paper.

***Normal Supergroup**. Term coined for the inverse concept of normality; a group of which the group under consideration is a normal subgroup.

***Opposites** or **Opposite Pairs**. See notes under group [4.03].

The ***Signature** of a group; see next section, under Classification.

A **Transitive** permutation group is one in which all bells permute to all positions, i.e. they are not partitioned in sets of positions. The overall partition of working bells must not be confused with the set of transposition cycles within a particular element of the group.

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