is the other perfect group of order 7200 i.e the direct product of two copies of SL(2,5) with amalgamated central subgroups (denoted by P4 by Sandlöbes
5] ORDER 7200 GENERATING CLASSES: [3] MAXIMALS: [3] [4]
UNION OF: [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] >clear; > " >SL(2,7) times SL(2,7) >" >g=matrix(4,gf(7)); >g.genera: >x=mat(6,0,0,0:
> 5,6,0,0: > 0,0,5,4: > 0,0,5,0), >y=mat(6,4,0,0: > 4,4,0,0: > 0,0,1,5: > 0,0,0,1); >print order(g); 112896 > > print x~4*y*x'-3*y; 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 > print x ‘14; 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 > print x ‘3*y‘2*x*y'5*x*y‘2 ; 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 > print x ‘2*y‘2*x'-l*y~2*x"-l*y~2*x‘2*y‘-l*x‘-l*y* 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 > print (x'2*y‘3)‘3; 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 > print (x*y‘2*x‘-l*y‘-l)*3; 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 >clear; >g=lree(x,y) ; >g.relations: >x‘4*y*x‘-3*y,
>x“14,
>x'3*y‘2*x*y'5*x*y‘2 ,
>x‘2*y'2*x'-l*y'2*x‘-l*y'2*x‘2*y ‘-l*x*-l*y~-l, >(x‘2*y‘3)‘3,
>(x*y‘2*x'-l*y‘-l)‘3; >print todd coxeter(g,[x]); 8064 >clear; >g=free(x,y); > g .relations: >x‘4*y*x‘-3*y, >x‘14, >x‘3*y‘2*x*y‘5*x*y'2, >x‘2*y"2*x~-l*y'2*x~-l*y‘2*x'2*y‘-l*x~-l*y'-1; >ind = todd coxeter(g,[x]);
>print ind * 14; 112896 >clear; >g=free(x,y) ; >g.relations: >x‘4*y*x‘-3*y, >x‘3*y‘2*x*y‘5*x*y‘2 ; >print order(g); 112896 >clear; >bye; END OF RUN. 338.929 SECONDS
R e fe r e n c e s
D.G. Arrell, S. M anrai and M .F. W orboys (1982), A procedure for obtain in g simplified defining relations for a subgroup, in G ro u p s-S t A ndrew s 1981, C.M . C am pbell and E .F . R oberston (eds), London M ath. Soc. Lecture N ote Series 71, C am bridge University P ress, C am bridge, p p .155-159.
D.G. Arrell and E .F . R o b ertso n (1984), A m odified T odd-C oxeter alg o rith m , in C om putational Group Theory, M ichael A tkinson (ed.), A cadem ic Press, London, p p .27-32.
M .J. B eetham and C.M . C am pbell (1976), A note on th e T odd-C oxeter coset enum eration algorithm , Proc. E dinburgh M ath. Soc. 2 0 , 73-78.
F .R . Beyl (1973), T he Schur m u ltip licato r of m etacyclic groups, Proc. A m er. M ath. Soc. 40 , 413-418.
F .R . Beyl (1986), T he Schur m u ltip licato r of S L (2, Z / m Z ) and th e congruence subgroup property, M ath. Z. 1 9 1 , 23-42.
F. R udolf Beyl and Jü rg en T appe (1982), G roup extensions, representations, and the Schur m ultiplicator, Lecture Notes in M ath em atics, 9 5 8 , Springer- Verlag, Berlin, Heidelberg, New York.
W .W . Boone (1955), C ertain sim ple unsolvable problem s of group theory, Indag. M ath. 16, 231-237, 492-497; 17, 252-256; 1 9 , 22-27,227-232.
W . B urnside (1911), T h eo ry o f groups o f finite order, C am bridge 1911, Dover Publishing 1955.
C.M. C am pbell, T . K aw am ata, I.M iyam oto, E .F . R o b ertso n an d P.D . W illiams (1986), Deficiency zero p resen tatio n s for certain perfect groups, Proc. Roy. Soc. Edinburgh 1 0 3 A , 63-71.
C.M. C am pbell and E .F . R o b ertso n (1978), Classes of groups related to F1a,&,c, Proc. Roy. Soc. Edinburgh 7 8 A , 209-218.
C.M. C am pbell and E .F . R o b ertso n (1980a), A deficiency zero p resen tatio n for S L ( 2 , p ) , Bull. London M a th . Soc. 12, 17-20.
C.M. C am pbell and E .F . R o b ertso n (1980b), O n 2-generator 2-relation soluble groups, Proc. E dinburgh M a th . Soc. 23, 269-273.
C.M. C am pbell and E .F . R o b ertso n (1982a), T he efficiency of sim ple groups of order < 105, C om m . Algebra 10, 217-225.
C M. C am pbell and E .F . R o b ertso n (1982b), G roups related to F a'b'cinvolv- ing Fibonacci num bers, in T he G eom etric Vein, C.Davis, B. G ru n b au m and F .A .S herk (eds.), Springer-Verlag, p p .569-576.
C M. C am pbell and E .F . R o b ertso n (1983), Some problem s in group presen ta tio n s, J. Korean M ath. Soc. 1 9
,
59-64.C.M . C am pbell and E .F . R o b ertso n (1984a), On th e F a'b'c conjecture, M itt. M a th . Sem . Giessen 1 6 4
,
25 - 36.C.M . C am pbell and E .F . R o b ertso n (1984b), P resen tatio n s for th e simple groups G, 105 < \G\ < 106, C om m . Algebra 1 2
,
127-156.C.M . C am pbell and E .F . R o b ertso n (1984c), O n a class of groups related to S L (2 ,2 n ), in C om putational G roup T h eo ry, Michael A tkinson(ed.), Academic Press, London, pp.43-50.
C.M . C am pbell and E .F . R o b ertso n (1988), C om puting w ith finite simple groups and th eir covering groups, in C om puters in A lgebra, M .C. Tangora (ed.), M arcel Dekker, New York, p p .63-71.
C.M . C am pbell, E .F . R obertson an d R.M . T hom as (1987a), O n groups related to Fibonacci groups, U niversity o f Leicester D ep a rtm en t o f C o m p u tin g Studies
Technical R eport 2
.
C.M . C am pbell, E .F . R o b ertso n an d R.M . T hom as (1987b), O n finite groups of deficiency zero related to (2, n)-g ro u p s, P a rt I, University of Leicester De p a rtm en t o f C om p u tin g Studies Technical R ep o rt 4
.
C.M . C am pbell, E .F . R o b ertso n and R.M . T hom as (1987c), On finite groups of deficiency zero related to (2, n)-g ro u p s, P a rt II, U niversity o f Leicester De p a rtm e n t o f C om p u tin g Stu d ies Technical R ep o rt 6
.
C.M . C am pbell, E .F . R o b ertso n and R.M . T hom as (1988), G roup p resen ta tions and sequences of Fibonacci ty p e, U niversity o f Leicester D ep a rtm en t o f
C o m p u tin g Studies Technical R ep o rt 8
.
C.M . C am pbell, E .F . R o b ertso n and P.D. W illiam s (1989), Efficient presen tatio n s for finite simple groups, in Groups - Korea 1988, A .C . Kim and B.H. N eum an (eds.), L ecture Notes in M ath em atics, 1 3 9 8
,
Springer-Verlag, Berlin Heidelberg New York, p p .65-72.C.M . C am pbell, E .F . R o b ertso n and P.D. W illiam s (1990a), On th e efficiency of some direct powers of groups, in Groups - Canberra 1989, L.G. Kovacs (ed.), Lecture Notes in M ath em atics, 1 4 5 6
,
Springer-Verlag, Berlin, Heidel-berg, New York, pp. 106-113.
C.M . C am pbell, E .F . R o b ertso n and P.D. W illiams (1990b), Efficient presen tatio n s of th e groups P S L ( 2 , p)
x
P S L{2, p), p prim e, J. London M a th . Soc. ( 2 ) 4 1 , 69-77.C.M. C am pbell and R.M . T hom as (1987), On (2 ,n )-g ro u p s related to F i bonacci groups, Israel J. o f M a th em a tics, 5 8 (3 ) , 370-380.
Jo h n J. C annon (1973), C o n stru ctio n of defining relations for finite groups, Discrete M ath. 5, 105-129.
Jo h n J. C annon (1984), An in tro d u ctio n to th e group theory language Cayley, in C om putational G roup T heory M ichael A tkinson (ed.), A cadem ic Press, London, p p .145-184.
Jo h n J. C annon, Lucien A. Dim ino, George Havas and Jan e W . W atson (1973), Im plem entation and analysis of th e T odd-C oxeter algorithm , M ath. C o m p u t. 27, 463-490.
H.S.M . Coxeter and W .J.O . M oser (1980), G enerators and relations for dis crete groups, Ergeb. M ath . 14, 4 th edition, Springer, Berlin, Heidelberg.
A. Dietze and M. Schaps (1974), D eterm ining subgroups of given finite index in a finitely presented group. Canad. J. M a th . 26, 769-782.
K .W . G ruenberg (1976), Relation m odules o f finite groups, Regional C onfer ence Series in M ath ., 25 , A m er. M ath . Soc., Providence, R .I..
George Havas (1976), C o m p u ter aided d eterm in atio n of a Fibonacci group, Bull. A ustral. M ath. Soc. 15, 73-79.
George Havas (1991), Coset en u m eratio n strategies, Technical R ep o rt No. 200, D epartm ent o f C o m p u ter Science, T h e U niversity o f Queensland.
George Havas and M .F. N ew m an (1983), M inim al p resen tatio n s for finite groups of prim e-pow er order, C om m . Algebra 1 1 ( 2 0 ) , 2267-2275.
George Havas, P.E. K enne, J.S. R ichardson and E .F . R o b ertso n (1984), A Tietze tran sfo rm atio n p ro g ram , in C om putational G roup Theory, M ichael A tkinson (ed.), A cadem ic P ress, London, p p .69-74.
George Havas and J.S . R ichardson (1983), G roups of exponent five an d class four, C om m . Algebra 11, 287-304.
George Havas, J.S. R ichardson and L.S. Sterling (1979), T he last of th e Fi bonacci groups, Proc. Roy. Soc. E dinburgh 8 3 A , 199-203.
George Havas and L.S. Sterling (1979), Integer m atrices and abelian groups, in Sym bolic and Algebraic C o m p u ta tio n, edited by E.W .N g. Lecture Notes in C o m p u ter Science, 7 2 , Springer, Berlin, p p .431-451.
O. Holder (1893), Die G ru p p en der O rdnung p 3, pq2, pqr, p 4. M ath. A n n . 43, 301-412.
B. H u p p ert (1967), Endliche G ruppen I, G rundlehren M ath . W issenschaften 1 3 4 , Springer, Berlin, Heidelberg, New York.
A. Jam ali (1988), C o m p u tin g w ith sim ple groups: m a xim a l subgroups and presentations, P h.D . Thesis, D ep artm en t of M athem atics, U niversity of St. Andrews.
A. Jam ali and E .F . R obertson (1989), Efficient p resen tatio n s for certain simple groups, C om m . Algebra 1 7 ( 1 0 ), 2521-2528.
R odney Jam es (1980), T he G roups of O rder p6 (p an O dd P rim e), M ath. C om put. 3 4 , 613-637.
D.L. Johnson (1990), Presentations o f groups, London M ath em atical Society S tudent Texts, 1 5 , C am bridge U niversity Press, C am bridge.
D.L. Johnson and H. M awdesley (1975), Some groups of Fibonacci type, J. A ustral. M a th . Soc. 2 0 , 199-204.
D.L. Johnson and E .F . R o b ertso n (1978), F inite groups of deficiency zero, in Homological Group T h eo ry, C .T .C Wall (ed.), London M ath em atical Society Lecture Note Series, 3 6 , C am bridge U niversity Press, C am bridge, p p .275-289.
R.D . K eane (1976), M inim al presen tatio n s of finite groups of order 3n for n < 6, unpublished m an u scrip t, U niversity of Adelaide.
P . E. K enne (1983), P resen tatio n s for some direct p ro d u cts of groups, Bull. Austral. M a th . Soc. 2 8 , 131-133.
P.E. K enne (1986), Efficient p resen tatio n s for th ree sim ple groups, C om m . Algebra 1 4 ( 5 ), 797-800.
P.E. K enne (1988), A new efficient soluble group, Technical R ep o rt T R -C S - 88-18, D ep a rtm en t o f C o m p u ter Science, A ustralian N ational University.
P.E. K enne (1990), Some new efficient soluble groups, C om m . Algebra 1 8 ( 8 ), 2747-2754.
D.E. K n u th and P. Bendix (1970), Simple word problem s in universal algebras, in C om putational Problem s in A b stra ct Algebra, J. Leech (ed.), Pergam on
P ress, p p .263-297.
J. Leech (1977), C o m p u ter proof of relations in groups, in Topics in group theory and co m p u ta tio n, M .P .J. C u rra n (ed.), Academ ic P ress, London, pp.38- 61.
D . H. M cLain (1977), An algorithm for determ ining defining relations of a subgroup, Glasgow M ath. J. 18, 51-56.
W . M agnus, A. K arrass and D. Solitar (1976), C om binatorial Group Theory, second revised edition, Dover, New York.
J.L . Mennicke (1959), Einige endliche G ru p p en m it drei Erzeugenden und drei R elationen, Arch. M a th . 10, 409-418.
J.L . Mennicke and B.H. N eum ann (1987), M ore on finite groups w ith few defining relations, Research R ep o rt 21, D ep a rtm en t o f M athem atics, In s titu te o f Advanced Studies, A ustralian N ational University.
J. N eubiiser (1967), Die U n terg ru p p en v erb än d e der G ru p p en der O rdnungen < 100 m it ausnahm e der O rdnungen 64 un d 96. H abilitationsschrift, Univ. Kiel.
J. N eubüser (1982), An elem entary in tro d u ctio n to coset-table m ethods in co m p u tatio n al group theory, in G ro u p s-S t A ndrew s 1981, C.M . C am pbell an d E . F. R oberston (eds), London M ath . Soc. L ecture Note Series 71, C am bridge U niversity P ress, C am bridge, pp.1-45.
B.H. N eum ann (1955), On some finite groups w ith trivial m ultiplicator, Publ. M ath. Debrecen 4, 190-194.
B.H. N eum ann (1985), Some finite groups w ith few defining relations, J. A u s tral. M ath. Soc., Ser. A 3 8 , 230-240.
B.H. N eum ann (1987), Yet m ore on finite groups w ith few defining relations, Research R ep o rt 31, D ep a rtm en t o f M ath em a tics, In s titu te o f A dvanced S tu d ies, A ustralian N ational University.
M .F. N ewm an and E.A . O ’Brien (19xx), A co m p u ter aided analysis of some finitely presented groups, J. A ustral. M ath. Soc., to ap p ear.
A. Niemeyer, W . Nickel an d M. Schonert (1988), G A P G etting sta rted and Reference m anual, Aachen.
P S. Novikov (1955), O n th e algorithm ic unsolvability of th e word problem in group theory, Trudy M at. In st, im Steklov No. 44., 143 pp. Izd a t. A kad. N auk. S S S R , Moscow.
Michael J. Post (1978), T hree g en erato r groups w ith zero deficiency, C om m . Algebra 6 ( 1 3 ) , 1289-1296.
E .F . R obertson (1980), A com m ent on finite nilpotent groups of deficiency zero, C anad.M ath. Bull. 2 3 ( 3 ) , 313-316.
E .F . R obertson (1982), Efficiency of finite sim ple groups an d th eir covering groups, in Finite groups - com ing o f age (C oncordia, M ontreal, 1982),C o n tem porary M athem atics, 4 5 , A m erican M ath em atical Society, Providence, R .I., p p .287-293.
E .F . R obertson and K. R u th erfo rd (1991), A com p u ter proof of relations in a certain class of groups, Proc. Roy. Soc. E dinburgh 1 1 7 A , 109-114.
K. R u th erfo rd (1989), C om putational techniques applied to group presen ta tions, P h.D . Thesis, D ep artm en t of M ath em atics, University of St. A ndrew s.
G ü n ter Sandlöbes (1981), Perfect groups of order less th a n 104, C om m . A lge bra 9, 477-490.
T.W . Sag and J.W . W am sley (1973), M inim al presentations for groups of order 27!,n < 6, J. A ustral. M a th . Soc. 15, 461-469.
I. Schur (1904), Ü ber die D arstellung der endlichen G ru p p en durch gebrochene lineare S u b stitu tio n en , J. Reine A ngew . M ath. 127, 20-50.
I. Schur (1907), U ntersuchungen ü b er die D arstellung der endlichen G ru p p en durch gebrochene lineare S u b stitu tio n en , J. Reine Angew . M a th . 1 3 9 , 85-137.
C.C. Sims (1987), Verifying nilpotence, J. Sym bolic C om putation 3, 231-247.
J . G. Sunday (1972), P resen tatio n s of th e groups S L ( 2 , m ) and P S L ( 2 , m ) , Canad. J. M ath. 2 4 , 1129-1131.
M. Suzuki (1982), Group T h eo ry I, G ru n d leh ren M ath. W issenschaften 2 4 7 , Springer, Berlin, Heidelberg, New York.
R ichard G. Swan (1965), M inim al resolutions for finite groups, Topology 4, 193-208.
J.A . T odd and H.S.M . C oxeter (1936), A practical m eth o d for en u m eratin g cosets in an ab strac t finite group, Proc. Edinburgh M ath. Soc. (2)5, 25-36.
J.W . W am sley (1970), T h e deficiency of m etacyclic groups, Proc. A m er. M ath. Soc. 24, 724-726.
A. W estern (1899), G roups of order p 3q, Proc. London M a th . Soc. 30, 209-263.
J. W iegold (1982), T he Schur M ultiplier: An E lem entary A pproach, in G ro u p s-S t A ndrew s 1981, C.M . C am pbell and E .F . R o b ersto n (eds), London M ath. Soc. Lecture Note Series 7 1, C am bridge University P ress, C am bridge, pp. 137-154.
J. W iegold (1989), O n some groups w ith trivial m ultiplicator, Bull. A ustral. M ath. Soc. 4 0, 331-332.
L. Wos, R. O verbeek, E. Lusk an d J. Boyle (1984), A u to m a te d Reasoning Introduction and A pplications, Prentice Hall, Engelwood Cliffs, New Jersey.
L. Wos (1989), O T T E R : A reference m anual, Technical R ep o rt, M athem atics and C o m p u ter Science Division, A rgonne N ational L aboratories, A rgonne, Illinois.
H .J. Zassenhaus (1969), A p resen tatio n of th e groups P S L ( 2 , p ) w ith three defining relations, Canad. J. M a th . 2 1, 310-311.