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Generators and Relations for Discrete Groups, 1957
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CONTENTS (vi-vii)
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PREFACE (v-vi)When we began to consider the scope of this book, we envisaged a catalogue supplying at least one abstract definition for any finitely-generated group that the reader might propose. But we soon realized that more or less arbitrary restrictions are necessary, because interesting groups are so numerous. For permutation groups of degree 8 or less (i. e., subgroups of S8), the reader cannot do better than consult the tables of JOSEPHINE BURNS (1915), while keeping an eye open for misprints. Our own tables (on pages 134-143) deal with groups of low order, finite and infinite groups of congruent transformations, symmetric and alternating groups, linear fractional groups, and groups generated by reflections in real Euclidean space of any number of dimensions. The best substitute for a more extensive catalogue is the description (in Chapter 2) of a method whereby the reader can easily work out his own abstract definition for almost any given finite group. This method is sufficiently mechanical for the use of an electronic computer. There is also a topological method (Chapter 3),. suitable not only for groups of low order but also for some infinite groups. This involves choosing a set of generators, constructing a certain graph (the Cayley diagram or DEHNsche Gruppenbild), and embedding the graph into a surface. Cases in which the surface is a sphere or a plane are described in Chapter 4, where we obtain algebraically, and verify topologically, an abstract definition for each of the 17 space groups of two-dimensional crystallography. In Chapter 5, the fundamental groups of multiply-connected surfaces are exhibited as symmetry groups in the hyperbolic plane, the generators being translations or glide-reflections according as the surface is orientable or non-orientable. The next two chapters deal with special groups that have become famous for various reasons. In particular, certain generalizations of the polyhedral groups, scattered among the numerous papers of G. A. MILLER, are derived as members of a single family. The inclusion of a slightly different generalization in § 6.7 is justified by its unexpected connection with SHEPHARD'S regular complex polygons. Chapter 8 pursues BRAHANA's idea that any group generated by two elements, one of period 2, can be represented by a regular map or topological polyhedron. In Chapter 9 we prove that every finite group defined by relations of the form:
can be represented in Euclidean n-space as a group generated by reflections in n hyperplanes. Many well-known groups belong to this family. Some of them play an essential role in the theory of simple Lie groups. We wish to express our gratitude to Professor REINHOLD BAER for inviting us to undertake this work and for constructively criticizing certain parts of the manuscript. In the latter capacity we would extend our thanks also to Dr. PATRICK Du VAL, Professor IRVING REINER, Professor G. DE B. ROBINSON, Mr. F. A. SHERK, Dr. J. A. TODD and Professor A. W. TUCKER. We thank Mr J. F. PETRIE for two of the drawings: Figs. 4.2, 4.3; and we gratefully acknowledge the assistance of Mrs. BERYL MOSER in preparing the typescript. University of Toronto, H. S. M. Coxeter University of Saskatchewan, W. O. J. Moser February 1957 |
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BIBLIOGRAPHY (144-152)
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Index (152-155)
People Index
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Chapter 1 Cyclic, Dicyclic and Metacyclic Groups (1-11)[1] After briefly defining such fundamental concepts as generators, factor groups and direct products, we show how an automorphism of a given group enables us to adjoin a new element so as to obtain a larger group; e.g., the cyclic and non-cyclic groups of order 4 yield the quatern-ion group and the tetrahedral group, respectively. Observing that the standard treatises use the term metacyclic group in two distinct senses, we exhibit both kinds among the groups of order less than 32, whose simplest known abstract definitions are collected in Table 1. Opinions seem to be evenly divided as to whether products of group elements should be read from left to right or from right to left. We choose the former convention, so that, if A and B are transformations, A B signifies the transformation A followed by B. 1.1 Generators and relations. Certain elements S1, S2, . Sm, of a given discrete group 0, are called a set of generators if every element of 6 is expressible as a finite product of their powers (including negative powers). Such a group is conveniently denoted by the symbol {.51, 52, , Sm}. When m = 1, we have a cyclic group {S} C, , whose order q is the period of the single generator S. If q is finite, S satisfies the relation Sq= E, where E denotes the identity element. A set of relations gk(S1, S2, . .1 S7) =E (k = 1, 2, . . s), (1.11) satisfied by the generators of 0, is called an abstract definition of 6 if every relation satisfied by the generators is an algebraic consequence of these particular relations. For instance, if q is finite, SI= E is an abstract definition of Eq. It is important to remember that, in such a context, the relation Sq= E means that the period of S is exactly q, and not merely a divisor of q. This is sometimes expressed by saying that the relation is not merely "satisfied" but "fulfilled" (see MILLER, BLICHFELDT and DICKSON 1916, p. 143). Returning to the general group 0, defined by 1.11, let be another group whose abstract definition in terms of generators T1, T2, .. , T„ is given by the relations h1 (T1, T2, . T,,) = E (1 1, 2, . . t). (1.12) Ergebn. d. Mathem. N. F. H. 14, Coxeter u. Moser 1 [2] 1.2 Factor groupsThen it is known that a necessary and sufficient condition for 0 to be isomorphic to is the existence of relations 7-1= (SD S2, • • •• Sin) (1= 2, • • •,n), (1.13) Si= Si (Ti, T2, T.) (i = 1, 2, • • •, nt), (1.14) such that 1.11 and 1.13 together are algebraically equivalent to 1.12 and 1.14 together (CoxETER 1934b). For instance, Re= E S3= T2= S-1T S T E are two possible definitions for C6, since the relations 1?6= E, S - R4, T= R3 are equivalent to S3 T2- S-4T ST= E, R = ST. 1.2 Factor groups. Let 6' e-/ {R1, R2, . R.} be defined by the s r relations The correspondence gk(R1,R2, . . R„,) =E (k = 1, 2, . . . , s r). Ri (1 1, 2, . . . , m) defines a homomorphism of G (defined by 1.11) onto 0'. The elements gk(Si, S2, . " Sin) (k = s +1, . . s + r) (1.21) of 6 all correspond to the identity element gk (Ri, R2, . Rin) =E (k = s 1, ...,s r) of 0'. Hence the kernel of the homomorphism is the normal subgroup 92 {W-igk (Si, S2, . (k = s+1,...,s+r), where W runs through all the elements of 6. In fact, 92 is the smallest normal subgroup of 6 that contains the elements 1.21, and it follows that 0' a?: 0/9Z In other words, the effect of adding new relations to the abstract de-finition of a group 0, is to form a new group 0' which is a factor group of 0 (KuRoscH 1953, p. 76). In particular, the effect of adding to 1.11 the relations Sri sis,= E (i,:i = 1, 2, . . . , m) is to form the commutator quotient group of 6, which is the largest Abelian factor group of G. [3] 1.3 Direct products Every group with in generators is a factor group of the free group am, which has in generators and no relations (REIDEMEISTER 1932a, p. 31). Apart from some special considerations in § 7.3, p. 88, we shall not attempt to describe the modern development of the theory of free groups, which began with the remarkable theorem of NIELSEN (1921) and SCHREIER (1927) to the effect that every subgroup of a free group is free (see especially MAGNUS 1939, BAER 1945, M. HALL 1949, CHEN 1951, 1954, Fox 1953, 1954 and KUROSCH 1953, pp. 271-274). 1.3 Direct products. If two groups 6, defined by the respective sets of relations 1.11, 1.12, have no common element except E, and if all elements of 6 commute with those of sj, then the in n elements Si and Ti generate the direct product 6 x S) . Clearly, a sufficient abstract definition is provided by 1.11, 1.12, and Tri Si = E (i= 1 , . . . , in; j = 1, . . . , n). However, in many cases the number of generators may be reduced and the relations simplified. As an example, consider the cyclic groups and C21 defined by the respective relations S3-= E and T2= E . Their direct product C3 X C21 of order 6, has the abstract definition 1.16; but it is also generated by the single element R = S T and defined by the single relation 1.15, which shows that C3 X C2 ed. Co. More generally, the direct product of cyclic groups of orders q and r is an Abelian group Co X Cr, of order qr, which is cyclic if q and I are coprime: Co x Cr C„ r, (q,r) = 1. Still more generally, if p, q, . are distinct primes, any Abelian group of order pacq13. is a direct product x 6oft x • • • of Abelian p-groups (BuRNsiDE 1911, pp. 100 107), and every such p-group is a direct product of cyclic groups: 62,« Cvoci x C„ces X • • • , where cq+ (x2+ • • • • This p-group is described as the Abelian group of order pa and type (ay a2, .); in particular, the direct product of a cyclic groups of order p is the Abelian group of order pc' and type (1, 1, . . . , 1) : c; c, x c, x • • x c, . [4] 1.3 ExamplesCombining the above results, we see that every finite Abelian group is a direct product of cyclic groups. The infinite cyclic group C„, is generated by a single element X. without any relations. Thus it is the same as the free group a, on one generator. The inverse Xs' is the only other element that will serve as a generator. The direct product Cc20 C Ca, of two infinite cyclic groups is defined by the single relation Y = YX . (1.31) Its finite factor groups are obtained by adding relations of the type Xb ye = E For example, in the Abelian group _go yc= AT-c yo = E, xy = YX, (1.32) we have xc= Yb, xc2._ ybc X-b2 and therefore Xn = E, where it -= b2 + c2. Suppose (b, c) = d Then Xd is a power of Y, namely Xd _ Xyb-flc = y-(13b-Hy c) Also b /9 c. Yb xc = y-c@b+yOld = Yb-ynld ye = = yb(tbl-vc)Id = ye-Ffinld Since (16, y) - 1, the period of Y divides n/d, and any element of the group is expressible as XxYv (0 x < d, 0 y<n/d). Consider the direct product C'd XCn/din the fowl Zd= Ynid= E, Z Y = YZ The element X - ZY-ob-Fve)ld satisfies XY = YX and Xb yc = Zb y{-b(flb+ye)+c(vb-Pc)}Id = Zb y-flida = E x-c yb = Z-c yfc((3b+yc)+b(yb-fic)}1d Z-c yynld E Hence our original group {X, Y} is CdX C„id, the direct product of cyclic groups generated by X yob+yeva and Y. It can be shown similarly that the Abelian group %ye, yb_ x-b y-c, y = yx or Ye= Zb, Ze= Xe- Yb, X YZ - ZY X = E (1.33) [5] 1.4 Automorphismsis Ca X cid, the direct product of cyclic groups generated by X Y(911-1-vc)id and Y, where t = b2+ bc c2 and d = (b, c) = y b -13 c (FRUcHT 1955, p. 12). 1.4 Automorphisms. Consider again the group 0 .2-4. {S1, S2, , S defined by 1.11. Suppose it contains nt elements Si, S2, . . S,„ which satisfy the same relations: S2, , S,'„) E (k 1, 2, . . s). Then the correspondence Si Si (i= 1, 2, . . m) (1.41) defines an automorphism of 6 if the elements Si are a set of generators, i.e., if every Si can be expressed in terms of them. One fruitful method for deriving a larger group 6* from a given group 0 is to adjoin a new element T, of period ac (say), which trans-forms the elements of 6 according to an automorphism of period c. If we identify P with an element U of period a in the centre of 6, the order of 0* is evidently c times that of 6 If the automorphism is given by 1.41, the larger group is defined by the relations 1.11 and T-1Si T = Si , Tc = U (1.42) This procedure is easily adapted to infinite groups. Although a may be infinite, 6 is still a normal subgroup of index c in 6*. In the case of an inner automorphism, 6 contains an element R such that, for every S in 6, R-1 S R = T-1 S T, i.e., TR-1 S = S T R-1. Thus the element Z = T R-1= R--1T of 6* commutes with every element of 6. The lowest power of Z that belongs to 6 is V - Zc= U of period b, say. V, like Z, commutes with every element of 6; since it belongs to 6, it belongs to the centre. If (b, c) = 1 (for instance, if b is prime to the order of the centre, as in COXETER 1939, p. 90), consider integers /3, y, such that yb fic=1. Instead of adjoining T to 6, we could just as well adjoin Z = T or adjoin Z Vfl zi +ft _ zvb [6] 1.5 Dihedral groupswhose cth power is Zyb c Vvb = E (since Vb- E). Hence in this case 0* "i 6 x , where is the cyclic group generated by Zvb. 1.5 Some well-known finite groups. The cyclic group C„, defined by the single relation (1.43) So= E, admits an outer automorphism of period 2 with transforms every element into its inverse. Adjoining a new element R1, of the same period, which transforms C, according to this automorphism, we obtain the dihedral group 0,, of order 2q, defined by 1.51 and that is, S Ri E , SI -= Ri = (SRi)2 - E . (1.52) The same group 0, is equally well generated by the elements R1 and R2= R1S, in terms of which its abstract definition is RT.= fq= (R1R2)q= E . (1.53) The "even" dihedral group 02m, defined by 1.53 with q - 2m, has a centre of order 2 generated by Z = (R1R2)m. If m is odd, the two elements R1 and R = R2Z satisfy the relations Ri= R2= (RiR)m E , so that {R1, R} is 07,2i and we have °2 m X °771 Since 02„, (m odd) can be derived from 0,, by adjoining R2, which transforms 0„, in the same manner as R, we see that 1.54 is an example of 1.43 (with a = b= 1, c= 2, f3= y = - T= R2, U V = E). When in = 1, 1.54 is the four-group (m odd). (1.54) defined by 2 1-4-1C2 X 4°1 X Ri - R2 = (R1R 2)2 = E . In terms of the three generators R1, R2 and Ro= R1R2, these relations become R(2) = - RoRiR2= E . (1.55) [7] 1.6 Dicyclic groupsIn this form, clearly admits an outer automorphism of period 3 which cyclically permutes the three generators. Adjoining a new element S which transforms?2in this manner, we obtain a group of order 12 defined by 1.55 and S3 = E , S-iRoSi = Ri (i = 1, 2). The same group is generated by S and Ro, in terms of which it has the abstract definition (1.56) Since the permutations S = (1 2 3) and Ro= (1 2) (3 4) generate the alternating group 214 of order 12 and satisfy the relations 1.56, we con-clude that the group defined by these relations is 214. The above deri-vation shows that 214 contains as a normal subgroup. 214 is equally well generated by S and U = S-1 R0, in terms of which its definition is S3= U3 (S U)2 E . (1.57) Clearly, 214 admits an outer automorphism of period 2 which inter-changes the generators S and U. Adjoining such an element T, we obtain a group of order 24 defined by 1.57 and T2= E, TS T= U. (1.58) In terms of the generators S and T, this group is defined by S3= T2= (S T)4= E (1.59) Since the permutations S = (2 3 4), T= (1 2), which generate the symmetric group e4 of order 24, satisfy 1.59, we conclude that the group defined by these relations is e4. In terms of the generators S and U = S-AT , s4 is defined by S3= U4= (S U)2 E . 1.6 Dicyclic groups. When q is even, say q -2m, the automorphism S 5-4 of q can be used another way. Adjoining to S2m= E (1.61) a new element T, of period 4, which transforms S into S-4 while its square is Sm, we obtain the dicyclic group <2, 2, m), of order 4m, defined by 1.61 and T2= sm, S T = Since the last relation may be written as (S T)2= T2, S and T satisfy Sm - T2- (S T)2. (1.62) [8] 1.7 The quaternion getupTo show that these two relations suffice to define <2, 2, nt>, we observe that they imply S T71 = T2= TA T2 7-, stn (T-1 S T)m S-?, which is 1.61 (CoxETER 1940c, p. 372; cf. MILLER, BLICHFELDT and DICKSON 1916, p. 62). In terms of the three generators S, T, and R --=-- S T, <2, 2, m> is defined by the relations R2= S m = T2= RS T, or in terms of R and T alone: R2_ T2 =. (R-1 y)n' (1.64) Of course, the symbol <2, 2, in> could just as well have been written as On, 2, 2> or <2, m, 2). Other groups <1, in, it) will be discussed in § 6.5. 1.7 The quaternion group. The smallest dicyclic group (2, 2, 2>, called the quaternion group, is defined by S2= T2 = (S T)2 or R2_ S2= T2= RS T. (1.71) Note the resemblance to the famous formula 12 = 72 k2 = k _ 1 of HAMILTON (1856, p. 446). 0 is the smallest Hamiltonian group, that is, it is the smallest non-Abelian group all of whose subgroups are normal. In fact, the Hamil-tonian groups are precisely the groups of the form (1.63) Ux21XCB, where 21 is an Abelian group of odd order, and 23 is an Abelian group of order 2m (in 0) and type (1, 1, . . 1) (HILTON 1908, p. 177; CAR-MICHAEL 1937, p. 114; ZASSENHAUS 1937, p. 123; SCORZA 1942, p. 89). is also the smallest group of rank 1, that is, it is the smallest non-Abelian group all of whose proper subgroups are Abelian. The groups of rank 1 have been investigated by MILLER and MORENO (1903), SCHMIDT (1924) and REDEI (1947). REDEI showed that, apart from 0, every such group belongs to one of three well-defined families. It thus appears that U is the only finite non-Abelian group all of whose proper subgroups are Abelian and normal. 1.8 Cyclic extensions of cyclic groups. If (q, r) = 1, the cyclic group 1.51 admits an automorphism S>- Sr, (1.81) 1.8 Cyclic extensions of cyclic groups [9] whose period c is the exponent to which r belongs modulo q, so that 1 (mod q) . We derive a group of order qc by adjoining a new element T, of period ac (where a divides both q and r- 1), such that T-'ST= Sr, To= Sala. q/a, we have the abstract definition Sm=Tc= U, Ua- E, T-4.5 T Sr (implying Ur = Srm = (TA ST) ?' = UT = U) for this group of order mac. (When c = 2 and r = - 1, the group is dihedral or dicyclic accor-ding as a = 1 or a = 2.) These relations can be simplified if (a,m) 1 . For if it a + a m = 1, we have S = a±am = SPa Ua = (Say Pa, so that 1.82 is generated by S1= Sa and T, in terms of which it has the abstract definition Ta° E T-1 SIT Si. Dropping the subscript, we are thus led to consider the group S'"= Pi E, T.-1S T = Sr, (1.83) Writing m of order mu, derived from C„, by adjoining T, of period n, which trans-forms Cm according to the automorphism 1.81, of period c. The new feature is that we no longer identify Te with an element of Cm. Since Srn = Tan S Tn= S, the consistency of the relations 1.83 requires rn.= 1 (mod m) (1.84) i.e., n must be a multiple of the exponent to which r belongs modulo in (CARMICHAEL 1937, p. 176). Thus 1.83 is a factor group of Tn = E, Ts-1S T..= Sr (1.85) (where r may be positive or negative). The group 1.85 is infinite if rn= 1, and of order n Irn-1 [10 ] 1.8 Cyclic extensions of cyclic groupsotherwise. When r - 1, 1.83 is obviously Cm x Cit. When r = is odd, 1.84 implies m = 2, so that 1.85 is C2x C, C27i• The centre of 1.83 evidently contains P, where c is the exponent to which r belongs modulo m. If 1 and n (a, c) = 1, where a - n/c, we can express T in terms of T1r- P and P, so that 1.83 is the direct product of Sm = T i = E, S Ti= 5r4 with the Ca generated by P. For instance, if a is odd, the group S" _ T2 a _ s T _ is the direct product (1.86) m Ca • (On the other hand, when in is odd and a = 2, 1.86 is (2, 2, m).) The period of S, being a divisor of r"- 1, may or may not be a multiple of r - 1. For a discussion of the former case, it is convenient to make a slight change of notation. In the group R(r--1) _ Tn E, T-1RT= Rr,(1.861) T transforms Rm into Rr m = R"'; thus Rm belongs to the centre. If (r - 1, in) = 1, (1.87) we can express R in terms of S = Rr-1 and Rm, so that in this case {R, T} is the direct product of 1.83 with the Cri. generated by Rm. For instance, when m is odd, the group R27"-T1- E, T-1 RT- R-1 is C2x <2, 2, m). We are thus led to consider the group 1.83 under the special con-dition 1.87. Since S is now a power of the commutator S T Sr = the commutator subgroup is the Cm generated by S. Its quotient group is defined by 1.83 and ST= TS, implying E = 5m, whence, by 1.87, S E; thus the commutator quotient group is the C„ defined by T" = E. [11] 1.8 Metacyclic groupsA group whose commutator subgroup and commutator quotient group are cyclic is conveniently called Z-metacyclic, i.e., metacyclic in the sense of ZASSENHAUS (1937, p. 138). Thus we have proved that, if 1.84 and 1.87 are satisfied, 1.83 is Z-metacyclic. ZASSENHAUS proved that, conversely, every finite Z-metacyclic group is expressible in this form. NEUMANN (1956, p. 191) has observed that this presentation of the general Z-metacyclic group is equivalent to the two relations where u is the inverse of r -1 modulo tn, so that u (r 1) = 1 (mod m). For instance, if m is odd, 1.86 is equivalent to Stn = T-4 S(n-1)12 T = Son+012. Let us call a Z-metacyclic group Z S-metacyclic if all its Sylow sub-groups are cyclic. For this, ZASSENHAUS (1937, p. 139) found the necessary and sufficient condition (m, n) = 1. (1.88) Thus the odd dihedral and odd dicyclic groups are Z S-metacyclic. But if (m, a) > 1, 0 mx C. (a odd) is only Z-metacyclic ; e.g., since 3 X C3 contains no element of period 9, its Sylow subgroup of order 9 is non-cyclic. The most important case of a Z S-metacyclic group is SP - T P-1 E , Sr, (1.89) where j5 is a prime and r is a primitive root (mod p). As NET-ro (1900, p. 284) attributes it to KRONECKER, let us call this K-tnetacyclic (see also HEFFTER 1898; MILLER, BLICHFELDT and DICKSON 1916, p. 12; CARMICHAEL 1937, pp. 42, 184). The term metacyclic was used in yet another sense by WEBER (1895, p. 598). 1.9 Groups of order less than 32. We have seen that the group 1.83 is C,„ x Cr, when r - 1, and that the cases when r = -1 (or m 1) include: X n/2 when it is odd (so that m = 2), when n/2 is odd (e.g., when it = 2), [12] 1.9 Groups of order less than 32<2, 2, m> when rn is odd and n - 4 , C2x <2, 2, m/2> when m/2 is odd and n =-- 4 Moreover, the same group with T inverted is obtained by changing r into r-1 (mod m). The remaining solutions of 1.84 with m n < 32 may be listed as follows: r -1 ± 9 4 4 5 5 4 4 3 3 2 2 2 2 2 2 4 5 7 9 8 8 15 15 12 12 In six of these cases (r^n -1) = m, so that the two relations 1.85 suffice. In two cases (r - 1, in) = (m, n) = 1, so that the group is Z S -meta-cyclic. One of these two cases is the K-metacyclic group 1.89 with p = 5. The rest of the entries in Table 1 are taken from various sources, such as CAYLEY (1889), BURNSIDE (1911, pp. 157-161), MILLER (1911), Miss BURNS (1915), MILLER, BLICHFELDT and DICKSON (1916, pp. 143 168), CARMICHAEL (1937, pp. 166-187), and COXETER (1939, pp. 81, 83, 143). For tables giving the number of groups of each order less than 162 (except 128), the reader may consult MILLER (1930) and SENIOR and LUNN (1934). For instance, the number of groups of order 160 (in-cluding Abelian groups) is 238. However, PHILIP HALL has informed us that MILLER was mistaken in giving the number of groups of order 64 as 294 ; the correct number is 267. |
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Chapter 2 Systematic Enumeration of Cosets[12] E. H. MOORE (1897) and many others have computed the index of a subgroup in an abstract group by systematically enumerating the cosets. TODD and COXETER (1936) converted this method into an almost mechanical technique, a useful tool with a wide range of applications. In § 2.1, we apply it to determine an abstract definition for a given finite group. In § 2.4, p. 17, we use it to find whether a given subgroup of an abstract group is normal. Finally, in § 2.5, we see how it helps in determining properties of new abstract groups. 2.1 Finding an abstract definition for a finite group. The principle of the method is as follows. Let us suppose that the relations g k(SS2, • .1 Sm) _E (k - 1, 2, . . . , s) (2.11) [13] 2.1 A subgroup and its cosetsare proposed as an abstract definition for a given group 6 of order g. In practice, these equations are obtained by taking a set of generators of 6 and observing some of the relations satisfied by them. It is natural to select relations of a simple form, such as those which express the periods of generators or of simple combinations of them. The number of relations to select is a matter for experiment, and the "method of cosets" is the way we test the success of the experiment. The group 6' defined by 2.11 either coincides with 6, or possesses a normal sub-group 61 such that the factor group 67061 is isomorphic to 6. In all cases the order of 6' is at least g. If we can verify that the order of 6' is at most g, it will follow that 6' is isomorphic to 6. To test whether this is or is not the case, we pick out a subset T,, T2, . . Tn of elements of 6' such that the relations 2.11 imply the relations hl (Ti, T2, . . . Tn) = E — 1, 2, , 1) (2.12) which are already known to be an abstract definition for a group 45' of order It < g. Conceivably the relations 2.11 imply other relations between the T's independent of 2.12, in which case the subgroup of 6' generated by the T's will not be 45' but some factor group• In all 1. cases the order of 4j is at most h. We now consider the sets of elements • R obtained by multiplying the elements of 4j on the right by the various elements of 6'. Two such sets either coincide or have no common member, and every element of 6' belongs to at least one such coset There will be at least g/h different cosets (otherwise the order of 6' would be less than h • g/h = g). If we can show that there are in fact exactly g/h distinct cosets, then the order of 6' will be at most It • g/h = g and it will follow that 6 and 6' are isomorphic. Let us expand each gic of 2.11 into a product of the generators and their inverses; e.g., (SI S22)2 will be expressed as ,S1 Si S1 Si'. S2-1 51 Si S1 S2-1 521 Such an expression, involving c factors st1, serves as the heading for a table of cosets having c 1 columns1). Each factor occurs between two adjacent columns (as on page 15). We denote the cosets (in an order to be defined soon) by the numerals 1, 2, 3, . . . . If two such numerals, a and 13, occur side by side in the pair of columns headed by Si 1, we understand that a • sp= 13. Thus, if ix is given, we can insert 13 = a • Sr', but if /3 is given, we can insert a = /3 • Sr'. The first and last columns of each table are identical. 1) TODD and COXETER (1936) used rows instead of columns because at that time they preferred to multiply the elements of a group from right to left. [14] 2.2 Setting up the tablesThe coset 1 is defined as the subgroup 45 and this information is put into every table. Thus, the information that we have to begin with is 1 • TJ— 1 --= 1, 2, . . . , n). Before proceeding we make sure that the number 1 appears in every table in every essentially different position. The coset 2 is defined as the set 1 • R where R is a suitably chosen element of 6' not belonging to Fj. This is done by inserting in one of the tables the number 2 in some (suitable) unoccupied place which is preceded or followed by the number 1. We now (and at all times) fill up as much as possible of all the tables before defining the next coset, in this case 3, and make sure that 2 appears in every essentially different place in every table before proceeding. When some new information is won, as happens when a row becomes complete, it is immediately put into the tables. When all the tables are complete, the process is at an end. If the number of cosets does not exceed g/h, our experiment has been successful: 6 and 6' are not merely homomorphic but isomorphic. 2.2 Examples. As an example of the enumeration method, consider the alternating group 215, of order 60, generated by the permutations S = (2 5 4), U = (1 2 3 4 5). Since S U — (1 2) (3 4), these generators satisfy the relations S3= Us= (S U)2= E, (2.21) which we propose as an abstract definition for 215. In order to show that the group 6' defined by 2.21 is of order 60, we take for the sub-group of order 5 generated by U. The tables with 1 • U = 1 inserted are SS S I U UUUU 1 1 I 1 1 1 1 1 1 S US U 1 1 1 Note that the symbol 1 (meaning .N appears in each table in every essentially different position. We define the coset 2 = 1 • S, and insert this into the tables to obtain S S S UUUUU S US U 1 2 1 1 1 1 1 1 9 Note that in the second and third tables row so that the tables contain 2 in every Define 3 — 2 • S. The first table yields 3 1 2 1 2 1 1 2 2 we have placed a 2 in a new essentially different position. • S = 1; then the third table [15] 2.2 The icosahedral groupyields 2 • U = 3. The tables become S s s uuuuu susu 1 2 3 1 1 1 1 1 1 1 2 3 Continuing in this manner, we define, in order, 4 = 3•U, 5 = 4 . S, 6 = 5•S, 7 = 4 • U, 8 7 . S, 9 = 8 . S, 10 = 9•U, 11 = 10 . S, 12=11'S. When the coset 12 has been defined the tables become 1 2 3 1 1 2 3 2 S S S 1 2 3 4 5 6 7 8 9 10 11 12 1 4 7 10 UUUUU S US U 1 1 1 1 1 1 1 2 3 1 1 2 3 4 7 5 2 2 3 4 5 2 6 9 10 11 8 6 6 4 7 8 6 12 12 12 12 12 12 9 7 5 6 9 10 11 8 9 10 12 10 11 12 12 The tables have "closed up" and hence the order of 6' is at most 60, as desired. Obviously, it is desirable to choose the subgroup as large as possible; but in an extreme case it may consist only of the identity element. It may happen that the proposed definition of 6 contains some relations that do not explicitly define the periods of the generators or of simple combinations of them, i.e., the relations are not of the form (Sci .522 ... ,S;71) P = E. For example, suppose we wish to show that the order of the quaternion group R2 = S2= T2— RST (2.22) is in fact 8. Before proceeding with the enumeration we could change these relations to the form Rz s-2_ s2 T-2 _ E; but it is not necessary to do so. We may simply set up our tables as RR I SS TT RST [16] 2.2 The quaternion groupand make sure that the four tables all have the same first column and all have the same last column. The tables for this group, with {E} as the subgroup, are: R R S 1 9 5 2 5 7 3 8 6 4 3 8 5 7 1 6 4 3 7 1 2 8 6 4 1 3 5 2 4 7 3 5 6 4 7 8 5 6 1 6 1 3 7 8 9 8 2 4 T T 1 4 5 2 6 7 3 2 6 4 5 8 5 8 1 6 7 3 7 3 2 8 1 4 R S T 1 9 4 9 5 6 3 8 2 4 3 5 5 7 8 6 4 7 7 1 3 8 6 1 5 7 6 8 1 3 2 4 Here the cosets have been defined as follows: 1 •R = 2, 1 • S = 3, T= 4, 2 • R -= 5, 2 • T 6, 5 • R 7, 3•R-n-- 8. The tables close up after the coset 8 has been entered, and hence the group defined by 2.22 is of order 8. In the course of the work it may happen that the information yielded by the tables implies that two of the numbers represent the same coset. (This may be the result of committing the error of not inserting all the available information into all of the tables at every stage.) In that case we replace the larger of the two numbers by the smaller throughout, and note any other coincidences which may result. When all these have been adjusted we may proceed in the usual manner. If such coincidences ultimately involve the collapse of the whole table (implying that every coset must be 1), then the subgroup is the whole group 6 (e. g , it may happen that the relations 2.11 imply S1= S2- • • • = Sm = E, so that 6 is of order 1). If the proposed relations are insufficient to define 6, the tables will fail to close up after g/h cosets have been introduced. Conversely, if the tables fail to close up when g/h cosets have been defined, then either the relations 2.11 are insufficient to determine 6, or else there are some undiscovered coincidences among the cosets already introduced. If the tables are "nearly" complete, it is probably the latter circumstance which has occurred, and the introduction of a few more cosets will demonstrate the coincidence. If on the other hand there are large gaps in the tables, it is a sign that the relations 2.11 are probably insufficient to define 6. The reader may like to test his skill by enumerating the five cosets of {V1, V2} in — V3 = V3= (V2V3)2 - (V3 V1)2 (V1 V2)2 E [17] 2.4 Normal subgroups(CARMICHAEL 1923, p. 255). Further examples may be found in TODD and COXETER (1936), COXETER (1939, p. 84), § 6.8, p. 81 and § 7.8, p. 99. 2.3 The corresponding permutations. The method yields a represen-tation of 6 as a transitive permutation group. Thus, in the first example S = (1 2 3) (4 5 6) (7 8 9) (10 11 12), U = (2 3 4 7 5) (6 9 10 11 8), and in the second example R= (1 2 5 7) (3 8 6 4), S = (1 3 5 6) (2 4 7 8), T (1 4 5 8) (2 6 7 3). However, the relation between 0 and the permutation group may well be only a homomorphism. In fact, it is a simple isomorphism only in the following cases: when {E}, so that the representation is regular, and when neither nor any proper subgroup of is normal in 6 (CARMICHAEL 1937, p. 157; see also BURNSIDE 1911, Chapters X to XII). 2.4 Finding whether a given subgroup is normal. As before, let {Ti, T2, . . . T7,} be a subgroup of index q in 6 {S1, S2, . . Sul} (defined by 2.11). Let denote the smallest normal subgroup of 6 containing $3. The factor group 6/.*, of order q* say, is defined by 2.11 and the further relations Ti= E (i 1, 2, . n) (2.41) obtained by equating the T's (which are expressed in terms of the S's and their inverses) to the identity element. It is apparent that is a normal subgroup if and only if $3*, or q = q*. Since it is usually easy to compute q and q* by the enumeration method, we have a useful tool for determining when a given subgroup is normal. The enumeration which determines q* is best carried out after the relations 2.11 and 2.41 have been "simplified". As a simple example, consider the group 0 214 defined by 9= T2= (S T)3= E. The subgroup {T, T S} is computed, by the enumeration method, to be of index 3. 6/* is defined by the relations S3= T2= (ST)3= E, T= E, — E, which are easily reduced to the single relation 9= E. Hence q* = 3, and is a normal subgroup. 2.5 How an abstract definition determines a group. Although the method is most often used in obtaining an abstract definition for a Ergebn. d. Mathew. N. F. H. 14, Coxeter u. Moser 2 [18 ] 2.5 An example of collapsegiven finite group, it may sometimes be used to determine the group defined by a given set of relations. For example, a natural generalization of the symmetric group 'OF in the form 1.59, is S3= (ST)'= E The order of the new group, and the indices of its subgroups, may be determined in this manner (cf. MILLER, BLICHFELDT and DICKSON 1916, p. 154). Similarly, the method may be used to show that certain relations are inconsistent. For example, consider the relations R S2= 53R, SR2= R3 S. (2.51) It is not an easy task to show, algebraically, that they imply (2.52) but this information is easily won by the method as follows. Choose the subgroup for the subgroup {R}. The tables with 1 • R — 1 inserted are S R R I RRRS 1 Define, in order, 2 = 1 • S, 3 — 2 • R, 4 — 2 • S, 5 = 3 • S. At this stage the tables are RSS SSSR S R 1? R R 1? S 1 2 3 4 5 1 9 3 5 2 4 4 1 2 3 4 5 2 4 5 4 4 1 2 3 4 5 2 4 5 3 4 2 5 4 1 2 3 4 5 1 3 2 1 2 3 5 4 1 3 2 2 5 4 We now define 6 = 5 • R and find, upon inserting this into the tables, that a complete collapse takes place, i. e., the tables imply 1 = 2 = 3 — 4 = 5 = 6. Since 1 • S = 2 = 1, it follow s that S is a power of R. With this information, the relations 2.51 yield 2.52 immediately. Chapter 3 Graphs, Maps and Cayley Diagrams The chief purpose of this chapter is to describe CAYLEY's representation of a group with given generators by a topological 1-complex or graph, whose vertices represent the elements of the group while certain sets of edges are associated with the generators. CAYLEY (1878 a, b) pro-posed the use of colours to distinguish the edges associated with different [19] 3.1 Graphsgenerators (see BURNSIDE 1911, pp. 423 427 and the frontispiece). Instead, for the sake of easier printing, we use lines drawn in various styles: ordinary, broken, dotted, etc After suitably embedding the graph into a surface, we obtain a map from which a set of defining relations for the group may be read off. The Cayley diagram was rediscovered by DEHN (1910, pp. 140-146). Accordingly, some authors (e.g., BAER and LEVI 1936, pp. 392, 393) call it the „DEHNsche Gruppenbild". But CAYLEY'S priority is indisputable, as he described it in the year of DEHN 'S birth. 3.1 Graphs. We assume that the reader is familiar with the idea of a finite or infinite graph, whose vertices (or nodes) are joined in pairs by directed edges (or branches). A graph is said to be connected if every pair of its vertices can be joined by a path along a set of consecutively adjacent edges. The same path described backwards is called the inverse path. A circuit is a path whose first and last vertices coincide. A trivial circuit consists of a path followed by its own inverse. A tree is a connected graph whose circuits are all trivial; its number of vertices is one more than the number of edges (KONIG 1936, p. 51). We regard a circuit as being essentially unchanged if it is combined with a trivial circuit; in particular, all trivial circuits are equivalent. With this understanding, any circuit may be regarded as including a certain "fixed" vertex E, and the product of two circuits is obtained by describing them in succession (REIDEMEISTER 1932a, p. 99). Any graph that is not a tree has a certain minimal set of fundamental circuits (KoNIG 1936, p. 61) such that every circuit of the graph is expressible as a finite product of their powers (including negative powers). Consider a connected graph having No vertices, NI edges, and y fundamental circuits. By removing an edge that belongs to only one of these circuits'), we obtain a new graph having No vertices, N1— 1 edges, and ,u — 1 fundamental circuits. Repeating the process, we obtain eventually a graph having No vertices, N1— it edges, and no fundamental circuits. Since this is a tree, No= N1-4u ± 1, that is, it N1 No -I- 1 (3.11) (KONIG 1936, p. 53). 3.2 MapsWe define a map to be the decomposition of an un-bounded surface into N2 non-overlapping simply-connected regions (called faces) by N1 arcs (called edges) joining pairs of No points (called vertices). It is understood that no two edges have a common interior 1) Any edge that belongs (non-trivially) to at least one fundamental circuit will serve. For if it belongs to more than one, we can replace all but one by their respective products with that one, to obtain a new set of fundamental circuits in which the given edge belongs to only one. 9* [20] itatifitir.",t1:7•:;;; point, and that every edge belongs to just two faces') (11 u.litiRT and Conn-VossEN 193'2, pp. 154 27t3; 13ALL 19:0, pp. 232 - 237; Lui SCHETZ 1949, pp. 72 -S5). The FAYLER•POINCARIt Chareidert?iie x No Art + (3.21) is a property of the surface, independent of the map; e.g., it is obviously the same for the dual map which has No faces, N, edges and Ni vertices. The possible values of x art: 2, 0, —4, . for an orientable surface, —2, 1, 0, —1, 2, . • • for a non-orientable surface. Each non-orientable surface can be derived from the corresponding orientable surface (called its two-sheeted covering surface) by suitably identifying pairs of points; e.g., the projective plane (or elliptic plane, x 1) can be derived from the sphere (x = 2) by identifying anti-podes. Any map drawn on the former corresponds to another map on the latter, having twice as many elements of each kind (THRELFALL 1932 a, pp. 41-42). The vertices and edges of a map evidently form a graph. Conversely (KONIG 1936, p. 198), any connected graph can be embedded into a surface so as to form a map having the same vertices and edges. The embedding is seldom unique, but there must be at least one for which N2, and therefore x, is maximal. When the maximal characteristic is 2, the graph is said to be planar (WHITNEY 1933) since it can be embedded into a sphere and so (say by stereographic projection) into a plane. The circuit round a simply-connected region formed by two or more adjacent faces is evidently equal to the product of circuits (in the same sense) around the faces themselves (see Fig. 3.2). In particular, when a planar graph is em-bedded into a plane, one of the NI faces appears as a peripheral circuit which, sur-rounding the remaining N2-1 faces, is equal to the product of circuits around them individually. In fact, these N2— 1 circuits are a fundamental system for the planar graph, so that velp-m¦so p 1•11---•• p ¦•¦••••• t Fig. 3,2. Products of circuits p = N2 1. 1) It may happen that the two faces coincide, i. e., that an edge occurs twice in the cycle of sides of one face. For instance, a tree forms a map on a sphere, having NI+ 1 vertices, NI edges, and one face (a 2 Nrgon). [21] 3 3 1:4001ny diarienim Comparing this with 3.11, we vet sly iiintitseii formula fur the sphere. 3.3 Cayley diagrams. CAYLEY (1878 a, b) represented the multi-plication table of a group 65, with given generators, by a graph having one vertex for each element of the group. It is convenient_ to use the same symbol V for an element anti its corresponding vertex. With each generator St, we associate a certain set of directed edges, my Si-edges, the direction being indicated by an arrow. Two vertices, V and IV, are joined by an St-edge, directed from V to 11 whenever') 11' omaSt ler (3.31) Thus at any vertex there are two edges for each generator, one directed towards the vertex and one directed away. The edges join the vertices to their "neighbours"; the neighbours of the vertex E represent the generators and their inverses. Any path along the edges of the graph corresponds to a word (an element expressed as a product of the generators and their inverses); e.g., the path P along an Se-edge forwards, then along an S3--edge backwards. then along an S1-edge forwards, corresponds to the word P.?1Sit S`. A path P leads from the vertex 1' to the vertex PI% If a word C satisfies the relation C E, then any path C is a circuit, and converse'],'; e.g.. if S E, then any path C Sr is a cyclically directed n-gon. Whenever a generator is invautory (i.e., of period two), the graph may be simplified by using a single undirected edge instead of the 2-circuit consisting of two oppositely directed edges joining the same two vertices, The permutations of the vertices which preserve the naming of the incident edges are precisely those given by the right regular represen-tation the of the group O. For if such a permutation takes the vertices V and W of 3.31 into the vertices V' and 11" respectively, then Si 1/1, Hence this permutation corresponds to the element U r 1.311 Vr".1 W 1) Like BURNSIDE (1911, p. 423), we adopt the original convention of CAYLKY (1878 b). But some authors would write I? l'Ss for which a precedent was provided by CAYLEY himself (1889). [22 ] which takes the vertex E and its neighbours St' into the vertex U and its neighbours S;i1U. The following are examples of Cayley diagrams for some well known groups of low order. For the alternating group 214, generated by the permutations S = (1 2 3), T.= (1 2) (3 4), the Cayley diagram can be drawn as in Fig. 3.3a (ICEmPE. 1886, p. 43). / \ S / I \ T - - - / \ I I \ I \ I I I Fig. 3.3 a. The tetrahedral group Fig. 3.3b. The quaternion group Readers interested in the Archimedean solids will recognize this as a Schlegel diagram for the truncated tetrahedron (SCHLEGEL 1883, pp. 353-358). For the quaternion group 0, in its regular representation S = (1256)(347 8), T= (2 3 6 7) (4 5 8 1), - - - % T.---.._ the Cayley diagram can be drawn as an octagon 1 2 3 4 5 6 7 8 with an inscribed octagram 1 4 7 2 5 8 3 6 (Fig. 3.3b). For the group of order 16 generated by the permutations R= (24) (3 6), S = (1 2)(3 5) (4 7) (6 8), T = (1 3) (2 8) (4 5) (6 7), we have an octagon joined to an oc-/ 7 V— tagram (Fig. 3.3c). For the group 03 r S3, generated by the transpositions R = (2 3), S = (31), T--z; (1 2), the Cayley diagram is the Thomsen graph (Fig. 3.3d). The cyclic Frig, 11.3c. A group of order 16 [23] 3.4 Planar Cayley diagramsgroup Ce , generated by R = (1 2 3 4 5 6), S = (1 4) (2 5) (3 6), is represented by the same graph with six edges directed, as in Fig. 3.3e. In each of the last two cases the generator S is redundant, and by deleting all the S-edges we obtain a simpler diagram for the same group with fewer generators. In general, a generator S, is redundant if and only if the deletion of all the Si-edges leaves a connected graph. In particular, every Cayley diagram for a given finite group is a sub-graph of the complete diagram (CAYLEY 1889) in which all the elements of the group (except E) are regarded as generators, and every vertex K.— p / \ / R / ; ¦% / / S - - - i \ i . / \ / / / / / 1 Fig. 3.3cL The non-cyclic group of order 6 Fig. 3..3e. The cyclic grrArp of order 6 of the graph is joined twice to every other. Though too unwieldy to be useful in practice, this complete diagram is very easy to describe: any two distinct vertices, V and IV, are joined by one edge directed from V to W and by another from IV to V; the former is marked !VV.', and the latter V 3.4 Planar diagrams We have seen that every circuit in a graph is expressible as a product of fundamental circuits (§ 3.1, p. 19), that the words in a group appear as paths in its Cayley diagram, that pro-ducts of words appear as products of paths, and that a word is equal to E if and only if the corresponding path is a circuit. It follows that a sufficient (though usually redundant) set of defining relations is = C2 = Co = E, (3.41) where the C's are words corresponding to a set of fundamental circuits, including (because of our artificial simplification) the word SI corre-sponding to any undirected Si-edge. In particular, if the Cayley diagram is a planar graph, we can embed it in a plane and use the N2— 1 visible faces (along with any undirected edges). A substantial reduction in the number of relations can be effected if the embedding happens to be symmetrical, i.e., if faces are taken into faces by the regular permutation group (Sr. In this case, if V1, [24 ] 3.5 Orientablc surfaces172, . . . lc are the vertices of a face, then for any element V, the vertices V1 I', V217, . . . , Vi, V form a face. Since Gr is transitive on the vertices, all of them are surrounded alike by faces; thus in writing out the re-lations 3.41, we may restrict attention to the faces at a single vertex. The Cayley diagram shown in Fig. 3.3 a is symmetrically embedded if we regard the outside region of the plane as one more (hexagonal) face, i. e., if we regard the figure not merely as a Schlegel diagram but as the surface of a solid truncated tetrahedron. The faces (and the undirected edge) at a vertex yield the relations S3= T2 (S T) (T S)3 --= E, one of which is obviously redundant. Thus 214 has the abstract definition .53— T2= (S T)3= E, in agreement with 1.56. When the Cayley diagram is symmetrically embedded into some surface other than a sphere (or plane), the faces at a vertex still yield true relations, but these generally do not suffice to define the given group. What further relations are needed ? Before answering this question, let us summarize the topological theory of unbounded surfaces. 15 Unbounded surfaces. In POINCARE'S theory, two directed cir-cuits on a given surface are considered to be equivalent if they are homotopic, i.e., if one can be continuously transformed into the other (LEFSCHETZ 1949, p. 159). The product of two circuits is obtained by distorting one of them (if necessary) until they have a common point, and then describing them in succession. In this sense, the classes of homotopic circuits are the elements of a group called the fundamental group of the surface (KEREKJARTO 1923, pp. 11, 178; HILBERT and COHN-VOSSEN 1932, p. 290). Its identity element is the class of circuits that bound simply-connected regions, i. e., of circuits that can be con-tinuously shrunk to a point. Thus the fundamental group of a sphere is the group of order 1. For a torus, or 'sphere with a handle" the fundamental group is the infinite group L x Coo defined by 1.31 or ab = ba or E; its two generators may be identified with the two circles that can be drawn in an obvious manner through an arbitrary point on the surface. For a surface of genus p, or sphere with p handles (KEREKJARTO 1923, pp. 151-157; THRELFALL 1932a, p. 33; BALL 1939, p. 233), it is the more general group • . aP bPP I) a-lb-1— E (3.51) [25] 3.5 Non-orientable surfaceswhich has a pair of generators for each handle. (This group is infinite for p z 1, since it has a factor group C. x Coo given by setting ak -=- bk-= E for every k > 1). A more symmetrical definition for the same group is AiA 2. . A2„Ar1 AI' • .. AtTri - E (3.52) (LEFSCHETZ 1949, p. 85). The above surfaces are orientable. The simplest non-orientable surface is the projective plane, which may be regarded as a sphere with antipodes identified, or as a hemisphere with opposite peripheral points identified, or as a sphere with a cross-cap (HILBERT and CoHN-VossEN 1932, p. 279). Its fundamental group is the C2 defined by A2 = E, whose single generator may be identified with a straight line of the projective plane. For a Klein bottle (or non-orientable torus, or sphere with two cross-caps) the group is 264 2‘4 - E (3.53) which is infinite since it has a factor group defined by Ai = A= E. For a sphere with q cross-caps (DEHN 1912, p. 131; LEFSCHETZ 1949, pp. 73-78) it is i122 =E, (3.54) with a generator for each cross-cap. (This is infinite for all q> 1, since an infinite factor group is obtained by setting Ak = E for k > 2.) After cutting an orientable or non-orientable surface along a set of m circuits (m, = 2p or q) which all pass through one point and generate the fundamental group, we can unfold it into a flat region bounded by a 2 m-gon whose sides, a b a b A Fig. 3.5 a. The torus and the projective plane in their natural order, represent the abstract definition of the funda-mental group in terms of these generators (LEFSCHETZ 1949, p. 165). For instance, the torus is unfolded into a rectangle, and the projective plane into a digon, as in Fig. 3.5a. (See also Fig. 3.5c for the cases p = 2 and q = 3.) By taking replicas of this polygon, one for each [26] 3.5 The universal covering surfaceelement of the fundamental group, and joining them along corresponding sides in the proper sense, we obtain the universal covering surface, which is simply-connected (THRELFALL 1932a, p. 8; HILBERT and COHN-VOSSEN 1932, p. 289). For instance, an infinity of rectangles fill the ordinary plane, as in Fig. 3.5 b, and two digons (regarded as hemispheres) fill the sphere. The above remarks show that, on the simply-connected covering surface, the sides of the polygons, marked as ge-nerators of the fundamental b group, form a Cayley diagram for that group. If we introduce a metric so as to be able to use congruent polygons with straight sides, we must face the fact that, since each vertex belongs to 2m 2m-gons, the sum of the angles of the polygon is just 2.m. This means that the appropriate plane is elliptic if q = 1, Euclidean if p = 1 or q = 2, and hyperbolic if p > 1 or q > 2. We have seen that the only finite fundamental groups are those of the sphere and the projective plane. This is the topological counterpart of the geometrical fact that the sphere and the elliptic plane have finite area, whereas the Euclidean and hyperbolic planes have infinite area. (For a remarkable drawing of the case p = 2, see HILBERT and COHN-VOSSEN 1932, p. 228, Abb. 249.) Fig. 3.5b. The fundamental group of the torus Fig. 3.5 c Along with the map of 2 m-gons on the universal covering surface, it is convenient to consider the dual map (on the same surface) which has a vertex in each face of the original map, an edge crossing each edge, and a face surrounding each vertex. The marking of edges can be carried over, and by directing them suitably we again have a Cayley
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August 24, 2024)