Chapter XII.
On the Graphical Representation of Group⁷³.

O ur discussions hitherto have been confined exclusively to groups of finite order. When however, as we now propose to do, we consider a group in relation to the operations that generate it, it becomes almost necessary to deal, incidentally at least, with groups whose order is not finite; for it is not possible to say a priori what must be the number and the nature of the relations between the given generating operations, which will ensure that the order of the resulting group is finite.

Many of the definitions given in respect of finite groups may obviously be extended at once to groups containing an infinite number of operations. Among these may be specially mentioned the definitions of a sub-group, of conjugate operations and sub-groups, of self-conjugate sub-groups, of the relation of isomorphism between two groups and of the factor-group given by this relation. In regard to the last of them, the isomorphism between two groups, one at least of which is not of finite order, may be such that to one operation of the one group there correspond an infinitely great number of operations of the other. On the other hand, all the results obtained for finite groups, which depend directly or indirectly on the order of the group, necessarily become meaningless when the group is not a group of finite order.

181. Suppose that

It will sometimes be convenient to avoid the use of negative indices in the expression of any operation of the group. To this end we may write

The group, thus defined and represented, is the most general group conceivable that is generated by $n$ distinct operations. Any two such groups, for which $n$ is the same, are simply isomorphic with each other.

Suppose now that

If $Ḡ$ is the group generated by these operations, an isomorphism may be established between $G$ and $Ḡ$ by taking ${\bar{S}}_i$ ($i=1,2,\dots ,n$) as the operation of $Ḡ$ that corresponds to the operation $S_i$ of $G$.

Then to every operation of $G$

If between the generating operations of $Ḡ$ there are several independent relations

182. We may at once extend the result of the preceding paragraph in the following way:—

THEOREM I. If $G$ is the group generated by the $n$ operations

In proving this theorem, it is sufficient to notice that, if we take ${\bar{S}}_i$ ($i=1,2,\dots ,n$) as the operation of $Ḡ$ which corresponds to the operation $S_i$ of $G$, then to each operation of $G$ a single definite operation of $Ḡ$ will correspond, while to the identical operation of $Ḡ$ there corresponds the self-conjugate sub-group $H$ of $G$.

The theorem just stated is of such a general nature that it is perhaps desirable to illustrate it by considering shortly some simple examples.

Let us take first the case of a group $G$, generated by two independent operations $S_1$ and $S_2$, subject to no relations; and let us suppose that the single relation

In respect of this sub-group, the operations of $G$ can be divided into an infinite number of classes of the form

For the operations of the class $S_1^p{S}_2^{q}H$, multiplied by those of the class $S_1^{p\prime }{S}_2^{q\prime }H$, give always operations of the class

As a second illustration, we will choose a case in which $Ḡ$ is of finite order. Let $G$ be generated by the operations $S$ and $T$, which satisfy the relations

Then $H$ is formed by combining in all possible ways the operations

Now it may be easily verified that in $G$, the operation $S{T}^2$ belongs to a set of three conjugate operations

The group $Ḡ$ will be defined by the laws according to which these $27$ classes combine among themselves; and the reader will have no difficulty in verifying that it is isomorphic with the non-Abelian group of order $27$, whose operations are all of order $3$ (§ 73).

Ex. If , $S_2$, …, $S_N$ are the operations of a group $G$ of finite order, prove that the totality of the operations of the form $S_p^{-1}{S}_q^{-1}{S}_p{S}_q$ generate a self-conjugate sub-group $H$; and that the group $\frac{G}{H}$ is an Abelian group. Shew also that, if $H\prime$ is a self-conjugate sub-group of $G$ and if $\frac{G}{H\prime }$ is Abelian, then $H\prime$ contains $H$. (Miller, Quarterly Journ. of Math., Vol. XXVIII, (1896), p. 266.)

183. For the further discussion of a group, as defined by its generating operations and the relations between them, a suitable graphical mode of representation becomes of the greatest assistance. To this we shall now proceed.

In the simple case in which the group is generated by a single unrestricted operation, such a representation may be constructed as follows. Let $C_1$ and $C_{-1}$ be two circles which touch each other; $C_2$ and $C_{-2}$ the inverses of $C_{-1}$ in $C_1$ and $C_1$ in $C_{-1}$; $C_3$ and $C_{-3}$ the inverses of $C_{-2}$ in $C_1$ and $C_2$ in $C_{-1}$, and so on. These circles (fig. 1) divide the plane in which they are

drawn into an infinite number of crescent-shaped spaces. Suppose now that the space between $C_1$ and $C_{-1}$ is left white, and the spaces between $C_1$ and $C_2$ and between $C_{-1}$ and $C_{-2}$ (on either side of this white space) are coloured black; the next pair on either side left white, the next coloured black, and so on. Then any white space may be transformed into any other (and any black into any other) by an even number of inversions at the circles $C_{-1}$ and $C_1$; and if $S$ denote the operation consisting of an inversion at $C_{-1}$ followed by an inversion at $C_1$, the space between $C_{-1}$ and $C_1$ will be transformed into another perfectly definite white space by the operation $S^n$, while conversely the operation necessary to transform the space between $C_{-1}$ and $C_1$ into any other given white space will be a definite power of $S$. Hence if one of the white spaces, say that between $C_{-1}$ and $C_1$, is taken to correspond to the identical operation, there is then a unique correspondence between the white spaces and the operations of the group generated by the unrestricted operation $S$; and the figure that has been constructed gives a graphical representation of the group. It should be noticed that the actual geometrical process of inversion, which has been here used to construct the spaces corresponding to the operations of the group, is in no way essential to the graphical representation. It is however convenient as giving definiteness to the construction; and later, when we deal with the case of a general group, such definiteness becomes almost a necessity.

In a precisely similar manner, the group generated by a single relation $S$, satisfying the relation

finite series of $n$ circles, each of which intersects the two next to it on either side at angles $\frac{\pi }{n}$, while the $n$ circles divide the plane into $2n$ spaces. If these are left white and coloured black in alternate succession, and if one of the white is taken to correspond to the identical operation, there is a unique correspondence between the white spaces and the operations of the group generated by $S$, where $S$ represents the result of successive inversions first at $C_{-1}$ and then at $C_1$.

This operation obviously satisfies the relation

The systems of circles in figures 1 and 2 have a common geometrical property which may be noticed here as it will be of use in the sequel. Successive inversions at any one of the pairs $C_{-1}$ and $C_1$, $C_1$ and $C_2$, $C_2$ and $C_3$ are equivalent to the operation $S$; and therefore successive inversions at $C_{-1}$ and $C_r$ are equivalent to the operation $S^r$. Hence the result of an even number of inversions at any of the circles in either figure is equivalent to some operation of the group that the figure represents.

184. We may now proceed to construct a graphical representation of the group which is generated by $n$ operations subject to no relations. To this end, suppose $n+1$ circles drawn, each of which is external to all the others while each touches two and only two of the rest. Such a system can be drawn in an infinite variety of ways: we will suppose, to give definiteness and simplicity to the resulting figure, that the $n+1$ points of contact lie on a circle, which cuts the $n+1$ circles orthogonally. If these $n+1$ points taken in order are $A_1$, $A_2$, …, $A_{n+1}$, the successive circles are $A_{n+1}{A}_1$, $A_1{A}_2$, …, $A_n{A}_{n+1}$. We will suppose that only so much of these circles is drawn as lies within the common orthogonal circle $A_1{A}_2\dots A_{n+1}$. The $n+1$ circular arcs $A_{n+1}{A}_1$, $A_1{A}_2$, … then bound a finite simply connected plane figure which we will denote provisionally by $P$. Suppose now that $P$ is inverted in each of its sides, that the resulting figures are inverted in each of their new sides, and so on continually. Then from their mode of formation no two of the figures thus arising can overlap either wholly or in part; and when the process is continued without limit, every point in the interior of the orthogonal circle $A_1{A}_2\dots A_{n+1}$ will lie in one and only one of the figures thus formed from $P$ by successive inversions.

If $P\prime$ is any one of the figures or polygons so formed, the set of inversions by which it is derived from $P$ is perfectly definite. For suppose, if possible, that $P\prime$ is derived from $P$ by two distinct sets of inversions represented by $\Sigma$ and $\Sigma \prime$. Then $\Sigma {\Sigma \prime }^{-1}$ is a set of inversions in the sides of $P$ which transforms $P$ into itself. But every set of inversions necessarily transforms $P$ into some polygon lying outside it, and therefore

with the set composing $\Sigma \prime$. It immediately follows that the polygons can be divided into two sets, according as they are derived from $P$ by an even or an odd number of inversions. The latter we shall suppose coloured black, and the former (including $P$) left white. Every white polygon will be surrounded by black polygons and vice versâ. Since there is only one definite set of inversions that will transform $P$ into any other white polygon $P\prime$, the $n+1$ corners of $P\prime$ will correspond one by one to the $n+1$ corners of $P$; and when the perimeters of the two polygons are described in the same direction of rotation with regard to their interiors, the angular points that correspond will occur in the same cyclical order. On the other hand, in order that the corresponding angular points of a white and a black polygon may occur in the same cyclical order, their perimeters must be described in opposite directions. In consequence of these results, we may complete our figure (fig. 3) by lettering every angular point of every polygon with the same letter that occurs at the corresponding angular point of the polygon $P$.

185. If now $T_1$, $T_2$, …, $T_{n+1}$ represent inversions at $A_1{A}_2$, $A_2{A}_3$, …, $A_{n+1}{A}_1$, the operation $T_{r-1}{T}_r$ leaves the corner $A_r$ of $P$ unchanged and it transforms $P$ into the next white polygon which has the corner $A_r$ in common with $P$, the direction of turning round $A_r$ coinciding with the direction $A_{n+1}{A}_n\dots A_1$ of describing the perimeter of $P$. For brevity, we shall describe this transformation of $P$ as a positive rotation round $A_r$. If then we denote the operation $T_{r-1}{T}_r$ by the single symbol $S_r$, we may say that $S_r$ produces a positive rotation of $P$ round $A_r$. Let $P_1$ be the new polygon so obtained; and let $P\prime$ be the polygon into which any other white polygon $P\prime$ is changed by a positive rotation round the corner of $P\prime$ that corresponds to $A_r$. Then if $\Sigma$ is the set of inversions that changes $P$ into $P\prime$, it also changes $P_1$ into $P\prime$: so that ${\Sigma }^{-1}{S}_r\Sigma$ changes $P\prime$ into $P\prime$, i.e. produces a positive rotation round the corner $A_r$ of $P\prime$; and $S_r\Sigma$ changes $P$ into $P\prime$.

Let us now represent the operations

Then every operation, consisting of a pair of inversions in the sides of $P$, can be represented in terms of

If $s>r$, this is of the form required. If $s<r$, the term $S_{n+1}$ that then occurs may be replaced by

Now it has been seen that no two operations, each consisting of a set of inversions in the sides of $P$, can be identical unless the component inversions are identical term for term. Hence no two reduced operations of the form

If then we take the polygon $P$ to correspond to the identical operation of the group $G$ generated by

Moreover, since the operation ${\Sigma }^{-1}{S}_r\Sigma$ is a positive rotation round the corner $A_r$ of the polygon $\Sigma$ (calling now $P$ the polygon $1$), a simple rule may be formulated for determining by a mere inspection of the figure what operation of the group any given white polygon corresponds to.

This rule may be stated as follows. Let a continuous line be drawn inside the orthogonal circle from a point in the white polygon $1$ to a point in any other white polygon, so that every consecutive pair of white polygons through which the line passes have a common corner, a positive rotation round which leads from the first to the second of the pair. This is always possible. Then if the common corners of each consecutive pair of white polygons through which the line passes, starting from $1$, are $A_p$, $A_q$, …, $A_r$, $A_s$, the final white polygon corresponds to the operation

186. The graphical representation of a general group we have thus arrived at is only one of an infinite number that could be constructed; and we choose this in preference to others mainly because the form of the figure and the relative positions of the successive polygons are readily apprehended by the eye. As regards the mere establishment of such a representation we might, still using the process of inversion for the purpose of forming a definite figure, have started with $n+1$ circles each exterior to and having no point in common with any of the others. Taking as the figure $P$ the space external to all the circles and inverting it continually in the circles, we should form a series of black and white spaces of which the latter would again give a complete picture of the group. It is however only necessary to begin the construction of such a figure in order to convince ourselves that it would not appeal to the eye in the same way as the figure actually chosen.

Moreover, as in the representation of a cyclical group, the process of inversion is in no way essential to the representation at which we arrive. Any arbitrary construction, which would give us the series of white and black polygons having, in the sense of the geometry of position, the same relative configuration as our actual figure, would serve our purpose equally well.

187. If $\Sigma$ is the operation which transforms $P$ into $P\prime$, and if $Q$ is the black polygon which has the side $A_r{A}_{r+1}$ in common with $P$, then $\Sigma$ transforms $Q$ into the black polygon which has in common with $P\prime$ the side corresponding to $A_r{A}_{r+1}$. If then we take $Q$ to correspond to the identical operation, any black polygon will correspond to the same operation as that white polygon with which it has the side $A_r{A}_{r+1}$ in common. In this way we may regard our figure as divided in a definite way into double-polygons, each of which represents a single operation of the group.

188. We have next to consider how, from the representation of a general group whose $n$ generating operations are subject to no relations, we may obtain the representation of a special group generated by $n$ operations connected by a series of relations

It has been seen (§ 181) that to the identical operation of the special group there corresponds a self-conjugate sub-group $H$ of the general group; or in other words, that the set of operations $\Sigma H$ of the general group give one and only one operation in the special group.

Hence, to obtain from our figure for the general group one that will apply to the special group, we must regard all the double-polygons of the set $\Sigma H$ as equivalent to each other; and if from each such set of double-polygons we choose one as a representative of the set, the totality of these representative polygons will have a unique correspondence with the operations of the special group.

We shall first shew that a set of representative double-polygons can always be chosen so as to form a single simply connected figure. Starting with the double-polygon, $P_1$, that corresponds to the identical operation of the general group as the one which shall correspond to the identical operation of the special group, we take as a representative of some other operation of the special group a double-polygon, $P_2$, which has a side in common with $P_1$. Next we take as a representative of some third operation of the special group a double-polygon which has a side in common with either $P_1$ or $P_2$; and we continue the choice of double-polygons in this way until it can be carried no further. The set of double-polygons thus arrived at of necessity forms a single simply connected figure $C$, bounded by circular arcs; and no two of the double-polygons belonging to it correspond to the same operation of the special group. Moreover, in $C$ there is one double-polygon corresponding to each operation of the special group. To shew this, let $C\prime$ be the figure formed by combining with $C$ every double-polygon which has a side in common with $C$; and form $C\prime \prime$ from $C\prime$, $C\prime \prime \prime$ from $C\prime \prime$, and so on, as $C\prime$ has been formed from $C$. From the construction of $C$ it follows that every polygon in $C\prime$ is equivalent, in respect of the special group, to some polygon in $C$. Similarly, every polygon in $C\prime \prime$ is equivalent to some polygon in $C\prime$ and therefore to some polygon in $C$; and so on. Hence finally, every polygon in the complete figure of the general group is equivalent to some polygon in $C$, in respect of the special group; and therefore, since no two polygons of $C$ are equivalent in respect of the special group, the figure $C$ is formed of a complete set of representative double-polygons for the special group.

Suppose now that $S$ is a double-polygon outside $C$, with a side $A\prime A\prime$ belonging to the boundary of $C$. Within $C$ there must be just one polygon, say $ST$, of the set $SH$. If this polygon lay entirely inside $C$, so as to have no side on the boundary of $C$, every polygon having a side in common with it would belong to $C$. Now since $S$ and $ST$ are equivalent, every polygon having a side in common with $S$ is equivalent to some polygon having a side in common with $ST$. Hence since $C$ contains no two equivalent polygons, $ST$ must have a side on the boundary of $C$; and if this side is $A\prime \prime A\prime \prime$, the operation $T$ of $H$ transforms $A\prime A\prime$ into $A\prime \prime A\prime \prime$. Moreover, no operation of $H$ can transform $A\prime A\prime$ into another side of $C$; for if this were possible, $C$ would contain two polygons equivalent to $S$. It is also clear that, regarded as sides of polygons within $C$, $A\prime A\prime$ and $A\prime \prime A\prime \prime$ belong to polygons of different colours. Hence a correspondence in pairs of the sides of $C$ is established: to each portion $A\prime A\prime$ of the boundary of $C$, which forms a side of a white (or black) polygon of $C$, there corresponds another definite portion $A\prime \prime A\prime \prime$, forming a side of a black (or white) polygon of $C$, such that a certain operation of $H$ and its inverse will change one into the other, while no other operation of $H$ will change either into any other portion of the boundary of $C$.

The system of double-polygons forming the figure $C$, and the correspondence of the sides of $C$ in pairs, will now give a complete graphical representation of the group. For the figure has been formed so that there is a unique correspondence between the white polygons of $C$ and the operations of the group, such that until we arrive at the boundary the previously obtained rule will apply; and when we arrive at a polygon on the boundary, the correspondence of the sides in pairs enables the process to be continued.

189. From the mode in which the figure $C$ has been formed, no two of the figures $CH$ can have a polygon in common, when for $H$ is taken in turn each operation of the self-conjugate sub-group $H$ of the general group $G$; also the complete set contains every double-polygon of our original figure. This set of figures, or rather the division of the original figure into this set, will then represent in a graphical form the self-conjugate sub-group $H$ of $G$. Moreover, the operations which transform corresponding pairs of sides of $C$ into each other will, when combined and repeated, clearly suffice to transform $C$ into any one of the figures $CH$ and will therefore form a set of generating operations of $H$.

190. A simple example, in which the process described in the preceding paragraphs is actually carried out, will help to familiarize the reader with the nature of the process and will also serve to introduce a further modification of our figure. The example we propose to consider is the special group with two generating operations which are connected by the relations

As a first step, we will take account only of the relation

Each double-polygon, except those of the set $S_1^m$, contributes two sides to the boundary of $C$, one belonging to a white polygon and

one to a black. The polygons, which border $C$ and have sides in common with $S_2^{\alpha }{S}_1^{\beta }$, are $S_1{S}_2^{\alpha }{S}_1^{\beta }$ and $S_1^{-1}{S}_2^{\alpha }{S}_1^{\beta }$; and these, regarded as operations of the special group, are equivalent to $S_2^{\alpha }{S}_1^{\beta +1}$ and $S_2^{\alpha }{S}_1^{\beta -1}$. Hence the correspondence between the sides of $C$ is such that

(i) to the side $A_1{A}_3$ of the white polygon $S_2^{\alpha }{S}_1^{\beta }$ corresponds the side $A_1{A}_3$ of the black polygon $S_2^{\alpha }{S}_1^{\beta -1}$;

(ii) to the side $A_1{A}_3$ of the black polygon $S_2^{\alpha }{S}_1^{\beta }$ corresponds the side $A_1{A}_3$ of the white polygon $S_2^{\alpha }{S}_1^{\beta +1}$.

When we now take account of the additional relations

In addition to the correspondences between the sides of $C$ to which those just written simplify when the indices of $S_1$ and $S_2$ are reduced , we have now also the correspondences, indicated in the figure by curved lines with arrow-heads, which result from the new relations. Our figure may be further modified in such a way that its form takes direct account of these four new correspondences. Thus without in any way altering the configuration of the double-polygons, from the point of view of geometry of position, we may continuously deform the figure so that the pairs of corresponding sides indicated by the curved arrow-heads are brought to actual coincidence. When this is done, the resulting figure will have the form shewn in fig. 6. The correspondence in pairs of the sides of the boundary is indicated in the figure by full and dotted lines.

The two unmarked portions $A_1{A}_3{A}_1$ correspond, as also do the two similar portions marked with a full line, and the two marked with a dotted line.

It will be noticed that, in this final form of the figure for the special group, direct account is taken of the finite order of the generating operations $S_1$ and $S_2$ and also of the operation $S_1{S}_2$. The simplification of the figure that results by thus taking account directly of the finite order of the generating operations, and the greater ease with which the eye follows this simplified representation, are immediately obvious on a comparison of figs. 5 and 6.

191. In the applications of this graphical representation of a group that we have specially in view, namely to groups of finite order, the generating operations themselves are necessarily of finite order. The generating operations

The simple example we have given makes it clear that, at least in some particular cases, relations of this form may be directly taken into account in constructing our figure; in such a way that in the complete figure, consisting of a finite or an infinite number of double-polygons, the correspondence in pairs of the sides of the boundary, if any, will depend upon further relations between the generating operations.

We may, in fact, always take account of relations of the form in question in the construction of our figure as follows.

Let us take as before $n+1$ arcs of circles

192. Exactly as with the general group, these polygons are coloured white or black according as they are derivable from $P$ by an even or an odd number of inversions. The corners of any white polygon correspond one by one to the corners of $P$; so that, when the perimeters of the polygons are described in the same direction, corresponding corners occur in the same cyclical order.

If now the operation of successive inversions at $A_{n+1}{A}_1$ and $A_1{A}_2$ is represented by $S_1$, and that of successive inversions at $A_{r-1}{A}_r$ and $A_r{A}_{r+1}$ by $S_r$, ($r=2,3,\dots ,n$); all operations, consisting of an even number of inversions in the sides of $P$, can be represented in terms of

Moreover, from the construction of the polygon $P$, these operations satisfy the relations

Again, if $P\prime$ is any white polygon of the figure, which can be derived from $P$ by the operation $\Sigma$, a positive rotation (§ 185) of $P\prime$ round its corner $A\prime$ is effected by the operation ${\Sigma }^{-1}{S}_r\Sigma$; and, if $P\prime \prime$ is the polygon so obtained, $P\prime \prime$ is derived from $P$ by the operation $S_r\Sigma$. It is to be observed that a positive rotation of a polygon round its $A_r$ corner is now an operation of finite order $m_r$.

Suppose now that two operations $\Sigma$ and $\Sigma \prime$ transform $P$ into the same polygon $P\prime$, so that $\Sigma {\Sigma \prime }^{-1}$ leaves $P$ unchanged. If this operation, written at length, is

193. The number of polygons in the figure and therefore the order of the group will still, in general, be infinite. We may now proceed, just as in the previous case of a quite general group, to derive from the figure representing the group $G$, generated by $n$ operations satisfying the relations

As has been seen in § 182, if $H$ is the self-conjugate sub-group of $G$ which is formed by combining all possible operations of the form

It is not now however necessarily the case that the figure $C$, as thus constructed, is simply connected. Let us suppose then that $C$ has one or more inner boundaries as well as an outer boundary, and denote one of these inner boundaries by $L$. If the sides of $L$ do not all correspond in pairs, and if $A\prime A\prime$ is a side of $L$ such that the other side $A\prime \prime A\prime \prime$ corresponding to it does not belong to $L$, we may replace the double-polygon $P\prime \prime$ in $C$ of which $A\prime \prime A\prime \prime$ is a side by the double-polygon, not previously belonging to $C$, of which $A\prime A\prime$ is a side. If $P\prime \prime$ has a side on the boundary $L$, the new figure $C\prime$ thus obtained will have one inner boundary less than $C$; and if $P\prime \prime$ has no side on the boundary $L$, the new inner boundary $L\prime$ that is thus formed from $L$ will contain one double-polygon less than $L$, while the number of inner boundaries is not increased. This process may be continued till the new inner boundary $L_1$ which replaces $L$ is such that all of its sides correspond in pairs.

Let now $A_s{A}_{s+1}$ and $A\prime A\prime$ be a pair of corresponding sides of $L_1$, such that $A_s{A}_{s+1}$ is transformed into $A\prime A\prime$ by an operation $h$ of the self-conjugate sub-group $H$. A side $A_t{A}_{t+1}$ of another boundary of $C$ may be chosen such that $A_s{A}_{s+1}$ and $A_t{A}_{t+1}$ are sides of a simply connected portion, say $B$, of $C$; while no side of $L_1$ except $A_s{A}_{s+1}$ forms part of the boundary of $B$. The polygons of $B$ are equivalent, in respect of the special group, to those of $Bh$. Moreover, since the sides of $L_1$ correspond in pairs, no side of $Bh$, except $A\prime A\prime$ can coincide with a side of $L_1$. Hence when $B$ is replaced by $Bh$, the inner boundary $L_1$ will be got rid of and no new inner boundary will be formed. Finally then, $C$ may always be chosen so as to form a single simply connected figure.

The simply connected plane figure $C$, which has thus been constructed, with the correspondence of the sides of its boundary in pairs, will now give a complete graphical representation of the special group. The rule already formulated will determine the operation of the group to which each white polygon corresponds; and when, in carrying out this rule, we come to a polygon on the boundary, the correspondence of the sides of the boundary in pairs will enable the process to be continued.

The correspondence of the sides of $C$ in pairs involves a correspondence of the corners in sets of two or more. Thus if $A_r$ is a corner of $C$ and if, of the $m_r$ white polygons which in the complete figure have a corner at $A_r$, $n_1$ lie within $C$, there must within $C$ be $m_r-n_1$ white polygons equivalent to the remainder, and each of these must have an $A_r$ corner on the boundary. If $A\prime$ is a corner of $C$ such that there are $n_2$ white polygons, lying within $C$ and having a corner at $A\prime$, and if one of the sides of the boundary with a corner at $A\prime$ corresponds to one of the sides of the boundary with a corner at $A_r$, these $n_2$ white polygons must be equivalent to $n_2$ of the white polygons, lying outside $C$ and having a corner at $A_r$. If

194. When the order of the group is finite, we may still further so modify our figure as to take account of the correspondence of the sides of the boundary in pairs. We may, in fact, by a suitable bending and stretching of the figure, bring corresponding sides of the boundary to actual coincidence. When this is done, the figure will no longer be a piece of a plane with a single boundary, but will form a continuous surface, which is unbounded and in general is multiply connected. Every point $A_r$ on the surface, which in the plane figure did not lie on the boundary, will be a corner common to $2m_r$ polygons alternately black and white; and, in consequence of what has just been seen in regard to the correspondence of corners of the boundary, the same is true for every point $A_r$ on the surface which in the plane figure consisted of a set of corresponding corners of the boundary. If $N$ is the order of the group, the continuous unbounded surface will be divided into $2N$ polygons, black and white. The configuration of the set of white polygons with respect to any one of them will, from the point of view of geometry of position, be the same as that with respect to any other; and the like is true for the black polygons. Such a division of a continuous unbounded surface is described as a regular division; and we have finally, as a graphical representation of any group of finite order $N$, a division of a continuous surface into $2N$ polygons, half black and half white, which is regular with respect to each

set. The correspondence between the operations of the group and the white polygons on the surface is given by the rule that a single positive rotation of the white polygon $\Sigma$ round its corner $A_r$ leads to the white polygon $S_r\Sigma$.

195. We may again here illustrate this final modification of the graphical representation of a finite group by a simple example. For this purpose, we choose the group defined by

This group (§ 74) is a non-Abelian group of order $8$, containing a single operation of order $2$. The reader will have no difficulty in verifying that the plane figure for this group is given by fig. 7; and that opposite sides of the octagonal boundary correspond. The single operation of order $2$ is

two holes through it. A view of one half of the surface divided into black and white triangles, is given in fig. 8. The half of the surface, not shewn, is divided up in a similar manner; and the operation of order $2$ replaces each triangle of one half by the corresponding triangle of the other, an operation which clearly leaves the six corners of the polygons undisplaced.

196. The form of the plane figure $C$, which with the correspondence of its bounding sides in pairs represents the group, is capable of indefinite modification by replacing individual polygons on the boundary by equivalent polygons. If however we reckon a pair of corresponding sides of the boundary as a single side and a set of corresponding corners of the boundary as a single corner, it is clear that, however the figure may be modified, the numbers of its corners, sides and polygons remain each constant. This may be immediately verified on replacing any single boundary polygon by its equivalent.

If now $A$ be the number of corners, and $E$ the number of sides in the figure $C$ when reckoned as above, $2N$ being the number of polygons, then the connectivity⁷⁶ $2p+1$ of the closed surface is given by the equation

When the group and its generating operations are given, the integer $p$ is independent of the form of the plane figure $C$, which as has been seen is capable of considerable modification. The plane figure $C$ however depends directly on the set of generating operations that is chosen for the group. For a given group of finite order, such a set is not in general unique; and the number of generating operations as well as their order will in general vary from one set to another. It does not necessarily follow, and in fact it is not generally the case, that the connectivity of the surface by whose regular division the group is represented, is independent of the choice of generating operations. There must however obviously be a lower limit to the number $p$ for any given group of finite order, whatever generating operations are chosen; this we shall call the genus of the group⁷⁷.

197. We shall now shew that there is a limit to the order of a group which can be represented by the regular division of a surface of given connectivity $2p+1$. If $N$ is the order of such a group, generated by the $n$ operations

the surface will be divided in $2N$ polygons of $n$ sides each. Let $A_1$, $A_2$, …, $A_n$ be the angular points of one of these polygons; and suppose that on the surface there are $C_1$ corners in the set to which $A_1$ belongs, $C_2$ in the set to which $A_2$ belongs, and so on. Round each corner $A_r$ there are $2m_r$ polygons; and each polygon has one and only one corner of the set to which $A_r$ belongs. Hence

Again, each side belongs to two and only to two polygons, so that the number of sides is

Using these values for $A$ and $E$ in the formula of § 196, we obtain the equation

A complete discussion of this equation for the cases $p=0$ and $p=1$ will be given in the next chapter.

When $p$ is a given integer greater than unity, we can determine the greatest value that is possible for $N$ by finding the least possible positive value of the expression

If $n>4$, this quantity is not less than $\frac{1}{2}$, since $m_r$ cannot be less than $2$.

If $n=4$, the simultaneous values

If $n=3$, we require the least positive value of

Now the three sets of values

each make $K$ zero; and therefore no positive value of $K$ can be less than the least of those given by

These sets of values give for $K$ the values $\frac{1}{12}$, $\frac{1}{20}$ and $\frac{1}{42}$. Hence finally, the absolutely least positive value of the expression is $\frac{1}{42}$, and therefore the greatest admissible value of $N$ is

Hence⁷⁸:—

THEOREM II. The order of a group, that can be represented by the regular division of a surface of connectivity $2p+1$, cannot exceed , $p$ being greater than unity.

198. If, when a group is represented by the regular division of an unbounded surface, we draw a line from any point inside the white polygon $1$ (or any other polygon) returning after any path on the surface to the point from which it started, it will represent a relation between the generating operations of the group. For in following out along the line so drawn the rule that determines the operation of the group corresponding to each white polygon, some operation

If the surface is simply connected, any such line can be continuously altered till it shrinks to a point; and therefore the $n+1$ relations between the $n$ generating operations completely define the group, since all other relations can be deduced from them.

If however the surface is of connectivity $2p+1$, there are $2p$ independent closed paths that can be drawn on the surface, no one of which can by continuous displacement either be shrunk up to a point or brought to coincidence with another; and every closed path on the surface can by continuous displacement either be brought to a point or to coincidence with a path constructed by combination and repetition of the $2p$ independent paths⁷⁹. Any one of these $2p$ independent paths will give a relation between the $n$ generating operations of the group, which cannot be deduced from the $n+1$ relations on which the angles of the polygons depend. Moreover, every relation between the generating operations can be represented by a closed path on the surface; and therefore there can be no further relation independent of the original relations and those obtained from the $2p$ independent paths. There cannot therefore be more than $2p$ independent relations between the $n$ generating operations of a group, in addition to the $n+1$ relations that give the order of the generating operations and of their product; $2p+1$ being the connectivity of the surface by whose regular division into $n$-sided polygons the group is represented.

The $2p$ relations given by $2p$ independent paths on the surface are not, however, necessarily independent. In fact we have already had an example to the contrary in § 195. On the closed surface, by the regular division of which the group there considered is represented, four independent closed paths can be drawn. Any three of the corresponding relations can be derived from the fourth by transformation.

The only known cases in which the $2p$ relations are independent are those of a class of groups of genus one (§ 205).

Ex. Draw the figure of the group generated by $S_1$, $S_2$, $S_3$, where

Shew from the figure that the special group, given by the additional relation

Note to § 194.

If in the process of bending and stretching, described in § 194, by means of which the plane figure $C$ is changed into an unbounded surface, the angles of the polygons all remain unaltered, the circles of the plane figure will become continuous curves on the surface. These curves on the surface, which we will still call circles, are necessarily re-entrant. It is not however necessarily the case that, on the surface, a circle will not cut itself.

In the plane figure for the general group, an inversion at any circle of the figure leaves the figure unchanged geometrically but interchanges the black and white polygons. Each circle is, in fact, a line of symmetry for the figure such that, in respect of it, there is corresponding to every white polygon a symmetric black polygon and vice versâ.

Similarly on the surface a circle which does not cut itself may be a line of symmetry, such that a reflection at it is an operation of order two which leaves the surface and its division into polygons unchanged, but interchanges black and white polygons. When this is the case, every circle on the surface will be a line of symmetry and no circle will cut itself. On the other hand no such operation can ever be connected with a circle which cuts itself.

When such lines of symmetry exist, Prof. Dyck speaks of the division of the surface as regular and symmetric.