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Hurwitz groups: A brief survey
Author:
Marston Conder
Journal:
Bull. Amer. Math. Soc. 23 (1990), 359-370
MSC (1985):
Primary 20F05; Secondary 57S25
MathSciNet review:
1041434
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References |
Similar Articles |
Additional Information
- 1.
Alan
F. Beardon, The geometry of discrete groups, Graduate Texts in
Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected
reprint of the 1983 original. MR 1393195
(97d:22011)
- 2.
Jeffrey
M. Cohen, On Hurwitz extensions by
𝑃𝑆𝐿₂(7), Math. Proc. Cambridge Philos.
Soc. 86 (1979), no. 3, 395–400. MR 542684
(80j:30069), http://dx.doi.org/10.1017/S0305004100056231
- 3.
Jeffrey
Cohen, On non-Hurwitz groups and noncongruence subgroups of the
modular group, Glasgow Math. J. 22 (1981),
no. 1, 1–7. MR 604811
(83c:20061), http://dx.doi.org/10.1017/S0017089500004419
- 4.
Marston
D. E. Conder, Generators for alternating and symmetric groups,
J. London Math. Soc. (2) 22 (1980), no. 1,
75–86. MR
579811 (81h:20002), http://dx.doi.org/10.1112/jlms/s2-22.1.75
- 5.
Marston
Conder, A family of Hurwitz groups with nontrivial centres,
Bull. Austral. Math. Soc. 33 (1986), no. 1,
123–130. MR
823860 (87e:20073b), http://dx.doi.org/10.1017/S000497270000294X
- 6.
Marston
Conder, Hurwitz groups with arbitrarily large centres, Bull.
London Math. Soc. 18 (1986), no. 3, 269–271. MR 829585
(87e:20073a), http://dx.doi.org/10.1112/blms/18.3.269
- 7.
Marston
Conder, The genus of compact Riemann surfaces with maximal
automorphism group, J. Algebra 108 (1987),
no. 1, 204–247. MR 887205
(88f:20063), http://dx.doi.org/10.1016/0021-8693(87)90135-9
- 8.
Marston
Conder and John
McKay, A necessary condition for transitivity of a finite
permutation group, Bull. London Math. Soc. 20 (1988),
no. 3, 235–238. MR 931183
(89b:20007), http://dx.doi.org/10.1112/blms/20.3.235
- 9.
H.
S. M. Coxeter, The abstract groups
𝐺^{𝑚,𝑛,𝑝}, Trans. Amer. Math. Soc. 45 (1939), no. 1, 73–150. MR
1501984, http://dx.doi.org/10.1090/S0002-9947-1939-1501984-9
- 10.
H.
S. M. Coxeter and W.
O. J. Moser, Generators and relations for discrete groups, 4th
ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in
Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin,
1980. MR
562913 (81a:20001)
- 11.
L.
Finkelstein and A.
Rudvalis, Maximal subgroups of the Hall-Janko-Wales group, J.
Algebra 24 (1973), 486–493. MR 0323889
(48 #2242)
- 12.
Daniel
Gorenstein, Finite groups, 2nd ed., Chelsea Publishing Co.,
New York, 1980. MR 569209
(81b:20002)
- 13.
G.
Higman, Construction of simple groups from character tables,
Finite simple groups (Proc. Instructional Conf., Oxford, 1969) Academic
Press, London, 1971, pp. 205–214. MR 0342603
(49 #7349)
- 14.
A.
Hurwitz, Ueber algebraische Gebilde mit eindeutigen
Transformationen in sich, Math. Ann. 41 (1892),
no. 3, 403–442 (German). MR
1510753, http://dx.doi.org/10.1007/BF01443420
- 15.
Gareth
A. Jones and David
Singerman, Complex functions, Cambridge University Press,
Cambridge, 1987. An algebraic and geometric viewpoint. MR 890746
(89b:30001)
- 16.
Peter
B. Kleidman, Richard
A. Parker, and Robert
A. Wilson, The maximal subgroups of the Fischer group
𝐹𝑖₂₃, J. London Math. Soc. (2)
39 (1989), no. 1, 89–101. MR 989922
(90e:20014), http://dx.doi.org/10.1112/jlms/s2-39.1.89
- 17.
John
Leech, Note on a conjecture of Coxeter, Proc. Glasgow Math.
Assoc. 5 (1960), 25–29 (1961) (French). MR 0150193
(27 #196)
- 18.
John
Leech, Generators for certain normal subgroups of (2,3,7),
Proc. Cambridge Philos. Soc. 61 (1965), 321–332. MR 0174621
(30 #4821)
- 19.
John
Leech, Note on the abstract group (2,3,7;9), Proc. Cambridge
Philos. Soc. 62 (1966), 7–10. MR 0190214
(32 #7628)
- 20.
Joseph
Lehner and Morris
Newman, On Riemann surfaces with maximal automorphism groups,
Glasgow Math. J. 8 (1967), 102–112. MR 0220924
(36 #3976)
- 21.
Joseph
Lehner, Lectures on modular forms, National Bureau of
Standards, Applied Mathematics Series, vol. 61, Superintendent of
Documents, U.S. Government Printing Office, Washington, D.C., 1969. MR 0264070
(41 #8666)
- 22.
S. A. Linton and R. A. Wilson, personal communication.
- 23.
A.
M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math.
Assoc. 5 (1961), 90–96 (1961). MR 0146724
(26 #4244)
- 24.
A.
M. Macbeath, Generators of the linear fractional groups,
Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967),
Amer. Math. Soc., Providence, R.I., 1969, pp. 14–32. MR 0262379
(41 #6987)
- 25.
J.
Neubüser, H.
Pahlings, and W.
Plesken, CAS; design and use of a system for the handling of
characters of finite groups, Computational group theory (Durham, 1982)
Academic Press, London, 1984, pp. 195–247. MR 760658
(86i:20004)
- 26.
Gerhard
Rosenberger, Von Untergruppen der Triangel-Gruppen, Illinois
J. Math. 22 (1978), no. 3, 404–413 (German). MR 497484
(81g:20092)
- 27.
Chih-han
Sah, Groups related to compact Riemann surfaces, Acta Math.
123 (1969), 13–42. MR 0251216
(40 #4447)
- 28.
D.
Singerman, Automorphisms of compact non-orientable Riemann
surfaces, Glasgow Math. J. 12 (1971), 50–59. MR 0296286
(45 #5347)
- 29.
A. Sinkov, On the group-defining relations (2, 3, 7; p), Ann. of Math. 38 (1937), 67-76.
- 30.
Andrew
J. Woldar, On Hurwitz generation and genus actions of sporadic
groups, Illinois J. Math. 33 (1989), no. 3,
416–437. MR
996351 (90i:20020)
- 31.
Andrew
J. Woldar, Sporadic simple groups which are Hurwitz, J.
Algebra 144 (1991), no. 2, 443–450. MR 1140615
(92k:20026), http://dx.doi.org/10.1016/0021-8693(91)90115-O
- 32.
M.
F. Worboys, Generators for the sporadic group
𝐶𝑜₃ as a (2,3,7) group, Proc. Edinburgh Math.
Soc. (2) 25 (1982), no. 1, 65–68. MR 648901
(83h:20023), http://dx.doi.org/10.1017/S0013091500004144
- 1.
- A. F. Beardon, The geometry of discrete groups, Springer-Verlag, Berlin, 1980. MR 1393195
- 2.
- J. M. Cohen, On Hurwitz extensions byPSL2(7), Math. Proc. Cambridge Philos. Soc. 86 (1979), 395-400. MR 542684
- 3.
- J. M. Cohen, On non-Hurwitz groups and non-congruence subgroups of the modular group, Glasgow Math. J. 22 (1981), 1-7. MR 604811
- 4.
- M. D. E. Conder, Generators for alternating and symmetric groups, J. London Math. Soc. (2) 22 (1980), 75-86. MR 579811
- 5.
- M. D. E. Conder, A family of Hurwitz groups with non-trivial centres, Bull. Austral. Math. Soc. 33 (1986), 123-130. MR 823860
- 6.
- M. D. E. Conder, Hurwitz groups with arbitrarily large centres, Bull. London Math. Soc. 18 (1986), 269-271. MR 829585
- 7.
- M. D. E. Conder, The genus of compact Riemann surfaces with maximal automorphism group, J. Algebra 108 (1987), 204-247. MR 887205
- 8.
- M. D. E. Conder and J. McKay, A necessary condition for transitivity of a finite permutation group, Bull. London Math. Soc. 20 (1988), 235-238. MR 931183
- 9.
- H. S. M. Coxeter, The abstract groups G, Trans. Amer. Math. Soc. 45 (1939), 73-150. MR 1501984
- 10.
- H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Springer-Verlag, Berlin, 1980. MR 562913
- 11.
- L. Finkelstein and A. Rudvalis, Maximal subgroups of the Hall-Janko-Wales group, J. Algebra 24 (1973), 486-493. MR 323889
- 12.
- D. Gorenstein, Finite groups, 2nd ed., Chelsea Publishing Company, New York, 1980. MR 569209
- 13.
- G. Higman, Construction of simple groups from character tables, Finite Simple Groups (M. B. Powell and G. Higman, eds.), Academic Press, New York, 1971, pp. 205-214. MR 342603
- 14.
- A. Hurwitz, Über algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893), 403-442. MR 1510753
- 15.
- G. A. Jones and D. Singerman, Complex functions: an algebraic and geometric viewpoint, Cambridge Univ. Press, 1987. MR 890746
- 16.
- P. B. Kleidman, R. A. Parker, and R. A. Wilson, The maximal subgroups of the Fischer groupFi23, J. London Math. Soc. (2) 39 (1989), 89-101. MR 989922
- 17.
- J. Leech and J. Mennicke, Note on a conjecture of Coxeter, Proc. Glasgow Math. Assoc. 5 (1961), 25-29. MR 150193
- 18.
- J. Leech, Generators for certain normal subgroups of (2, 3, 7), Proc. Cambridge Philos. Soc. 61 (1965), 321-332. MR 174621
- 19.
- J. Leech, Note on the abstract group (2, 3, 7; 9), Proc. Cambridge Philos. Soc. 62 (1966), 7-10. MR 190214
- 20.
- J. Lehner and M. Newman, On Riemann surfaces with maximal automorphism groups, Glasgow Math. J. 8 (1967), 102-112. MR 220924
- 21.
- J. Lehner, Lectures on modular forms, Appl. Math. Ser., vol. 61, U. S. National Bureau of Standards, Washington, D.C., 1969. MR 264070
- 22.
- S. A. Linton and R. A. Wilson, personal communication.
- 23.
- A. M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961), 90-96. MR 146724
- 24.
- A. M. Macbeath, Generators of the linear fractional groups, Number Theory, Proc. Sympos. Pure Math., vol. XII, Amer. Math. Soc. Providence, R.I., 1969, pp. 14-32. MR 262379
- 25.
- J. Neubüser, H. Pahlings, and W. Plesken, CAS: Design and use of a system for the handling of characters of finite groups, Computational Group Theory (M. Atkinson, ed.), Academic Press, New York, 1984, pp. 195-247. MR 760658
- 26.
- G. Rosenberger, Von Untergruppen der Triangel-Gruppen, Illinois J. Math. 22(1978), 403-413. MR 497484
- 27.
- C.-H. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13-42. MR 251216
- 28.
- D. Singerman, Automorphisms of compact non-orientable Riemann surfaces, Glasgow Math. J. 12 (1971), 50-59. MR 296286
- 29.
- A. Sinkov, On the group-defining relations (2, 3, 7; p), Ann. of Math. 38 (1937), 67-76.
- 30.
- A. J. Woldar, On Hurwitz generation and genus actions of sporadic groups, Illinois J. Math. 33 (1989), 416-437. MR 996351
- 31.
- A. J. Woldar, Sporadic simple groups which are Hurwitz, preprint. MR 1140615
- 32.
- M. F. Worboys, Generators for the sporadic group Co, Proc. Edinburgh Math. Soc. 25 (1982), 65-68. MR 648901
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0273-0979-1990-15933-6
PII:
S 0273-0979(1990)15933-6
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