THE COMPLETE ENUMERATION OF FINITE GROUPS OF THE FORM R 2 i ={R i R j ) k -i=i

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1 ENUMERATON OF FNTE GROUPS OF THE FORM R ( 2 = (RfR^'u =1. 21 THE COMPLETE ENUMERATON OF FNTE GROUPS OF THE FORM R 2 ={R R j ) k -= H. S. M. COXETER*. ths paper, we vestgate the abstract group defed by the relatos f order tofdwhat values of the tegers k (j wll make the group fte. f some of the relatos (2) are abset, we ca suppose the correspodg k's to be fte; but ths ever happes whe the group s fte. f ay k H = 1, Rj s merely a alteratve ame for?,-; therefore we may suppose that k h > 1. t s coveet to represet the group by a graph of dots ad lks, as the Table at the ed of ths paper. The dots represet the geerators. The umbers wrtte uder certa lks are values of k (j. Wheever a lk s ot so umbered, we uderstad that k u =. Wheever two dots are ot (drectly) lked, we uderstad that k u = 2. The group s sad to be rreducble or reducble accordg as ts graph s coected or dscoected. f reducble, t s the drect product of two or more rreducble groups, represeted by the coected peces of the graph. Wth the help of certa lemmas, we shall prove the followg THEOREM J. The oly rreducble fte groups of the form are, [»], [»,4], [k], [,5], [,4,], [,,5], ~ ~ > ", ~, > ",,, 1! h j,,, ", (The umbers occurrg these symbols are the values of those k's that are greater tha 2. For detals, see the Table.) * Receved 0 October, 19; read 10 November, 19. compact form, (?,2;)*'V = 1 (1 ^^j^m, k ; = 1). X Cf. H. S. M. Coxeter, Aals of Math., 5 (194), 601 (Theorem 9). We shall refer to ths paper as D.g.g.r.

2 22 H. S. M. COXETER LEMMA 1. // the partcularform where b 1 = b 2 =... = b p^b ( (p^l, = p+l : p+2,..., w) r v s varat uder a certa orthogoal substtuto, the a*, 2 s lkewse x varat. Sce the substtuto s orthogoal, Zx? must be varat, ad so also ' must the dfferece H (b j b 1 )x j 2. P-rl The equato 2 (b j b 1 )x 2 = 0 represets a (p tmes degeerate) coe, whose "vertex " s thep-space X p+1 = X pjr 2 =... = X = 0. Sce the coe s varat, ths js-space must also be varat. Therefore the form Sa;, 2 wll stll be varat whe ^e replace all save the frst p x's by zero. LEMMA 2. Ay lear trasformato tjat leaves varat the two partcular forms (1) s«,y,«(^=1), (2) c,2/, 2 (c,.>0 for all j), V also leaves varat Tyf. f 6 1? b 2,... b are real o-vashg umbers,, such that the form HbjX? s varat uder a certa orthogoal substtuto, the Lemma 1 shows that the sums of the postve ad egatve terms are separately varat. Therefore ' s varat. Sce a orthogoal substtuto s merely a lear trast formato that leaves Sar 2 varat, Lemma 2 ca ow be deduced by

3 ENUMERATON OF FNTE GROUPS OF THE FORM Rf = (?,? ; ) fc; > = 1. 2 puttg 6 j = c7 1 Abbrevato. For "group geerated by reflectos" we-wrte g.g.r. LEMMA. Every fte g.g.r. the geeralzed Mkowsha space* S s T { s smply somorphc wth a g.g.r. the Eucldea space *S S+/ Let the geeral pot of S s T' be (x v x 2,..., x s+t ), at dstace s+ from the org. t s well koav that ay fte lear group leaves varat a postve defte quadratc form, e.g. the sum of all trasforms of Sz, 2. Ths must hold, partcular, for a fte group of cogruet trasformatos S s T l. By a sutable chage of coordates, uder whch the jjeeral pot becomes {y x, y 2,...,y s+t ), at dstace from the (ew) org, the varat form becomes (say) t 1 8+t By Lemma 2, we ca ow assert the varace of 2 yf. We thus have a group of cogruet trasformatos S s T l, leavg t+t 2 yf varat. By gvg the varables y a ew geometrcal terpreta- to, we ca regard the same algebrac substtutos as cogruet s+t trasformatos S s+t, leavg 2 } y^ varat. A reflecto s characterzed by the fact that t leaves varat every pot whose coordates satsfy a certa lear equato. Therefore, reflectos rema reflectos whe we pass from S s T l to S s+t. * H.S.M.Ck>xeteradJ.A.Todd, Proc. Camb. PM.Soc., 0(194), 1-. The reflecto the prme la^xj = 0 s the trasformato where A =:»/= *, 2«,-o,-X { = 1, 2 s+t),

4 24 H. S. M. COXETER LEMMA 4. Every fte group of the form?, 2 = {R R ) k J= 1 ca be geerated by reflectos the boudg prmes of a sphercal smplex. We kow* that there exst, some geeralzed Mkowska space S* T' (5-r-/ <w), w-j-1 pots Jl? -4 1,..., -4 T ", such that A A = l (l< ^U> = 2 cos (w/2jt rt ) (1 < < j < m). Let a 1 ' (1 ^ ^ w) deote the prme, through A 1, perpedcular to A 0 A'. The t s easly see that a', a> are cled at a agle r/k^. The prmes a' cut off a certa rego aroud A 0. (f closed, ths rego s a polytope, ad A 0 s ts -cetre.) Reflecto ay prme a' gves a ew rego, cogruet f to the frst. Reflecto ay other boudg prme of the ew rego gves a thrd rego; ad so o. Sce all the dhedral agles are of the form rjk, the regos Trll ft together wthout overlappg, ad there wll be o terstces j. fact, the rego bouded by the prmes a 1 ' s a fudametal rego for the group geerated by the reflectos these prmes. Let?,- deote the reflecto the prme a'. The relatos (1) ad (2) evdetly hold. The R's, so defed, may perhaps satsfy other relatos, ot deducble from these. But we ca assert that the g.g.r. s smply somorphc ether wth the abstract group defed by (1) ad (2) or wth a factor group thereof. Hece, f the abstract group s fte, the g.g.r. s a fortor fte. By Lemma, the g.g.r. occurs Eucldea space. By Theorem 8 of Dscrete groups geerated by reflectos c, t s smply somorphc wth the whole abstract group. Sce the org s varat^, the group ca be regarded as operatg sphercal space. By Lemma 4. 7 of the paper just cted**, f the umber of dmesos s take as small as possble, the sphercal fudametal rego s a smplex. Our theorem ow follows from the eumerato of Groups whose fudametal regos are smplexes^. * Coxeter ad Todd, loc. ct., 1. * Or, rather, eatomorphows. : Cf. D.g.g.r., 596. The argumet used D.g.g.r. shows that such extra relatos wll appear oly f the part of space flled by the fudametal rego ad ts trasforms s multply-coected. : D.g.g.r., 599. f! Eucldea (or Mkowska) space, every fte group of cogruet trasformatos leaves varat the cetrod of all the trasforms of a pot of geeral posto. D.g.g.r., 597. ft Joural Lodo Math. Soc, 6 (191), For a fuller accout, see Proc. Lodo Math. Soc. (2), 4 (192), both these papers (the former, the peultmate le o p. 12; the latter, the last le o p. 1G), a t should be a,,.

5 ENUMERATON OF FNTE GROUPS OF THE FORM R? = (l? f?,)*^ = TABLE OF RREDUCBLE FNTE GROUPS OF THE FORM Jtf = («, «, )*«= 1. Symbol Graph Order* [ ] t [ *. 4] [, 5] [, 4, ] [,, 5] _ ",, _,, -, ,,,, Trty College, Cambrdge. * Proc. Lodo Math. Soc. (2), 4 (192), 160, 159. All the orders save the last three ca be deduced from the theory of regular polytopes. For the rest, see Phl. Tras. Royal Soc. (A), 229 (190), t The symmetrc group of degree m+1. [ ] s the group of order 2; ts graph s a sgle dot. m s the umber of dots the graph.e. the umber of geerators. X The dhedral group of order 21:. [] ad [4] have already occurred above. The exteded cosahedral group. ("1 s the same as [, ]. *j The group of the twety-seve les o the cubc surface.