Geodesic
The efficiency of PSL(2, p)3 and other direct products of groups
Glasgow mathematical journal, Tome 39 (1997) no. 3, pp. 259-268

Voir la notice de l'article provenant de la source Cambridge University Press

A finite group G is efficient if it has a presentation on n generators and n + m relations, where m is the minimal number of generators of the Schur multiplier M (G)of G. The deficiency of a presentation of G is r–n, where r is the number of relations and n the number of generators. The deficiency of G, def G, is the minimum deficiency over all finite presentations of G. Thus a group is efficient if def G = m. Both the problem of efficiency and the converse problem of inefficiency have received considerable attention recently; see for example [1], [3], [14] and [15]. 
Campbell, C. M.; Miyamoto, I.; Robertson, E. F.; Williams, P. D. The efficiency of PSL(2, p)3 and other direct products of groups. Glasgow mathematical journal, Tome 39 (1997) no. 3, pp. 259-268. doi: 10.1017/S0017089500032195
@article{10_1017_S0017089500032195,
     author = {Campbell, C. M. and Miyamoto, I. and Robertson, E. F. and Williams, P. D.},
     title = {The efficiency of {PSL(2,} p)3 and other direct products of groups},
     journal = {Glasgow mathematical journal},
     pages = {259--268},
     year = {1997},
     volume = {39},
     number = {3},
     doi = {10.1017/S0017089500032195},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500032195/}
}
TY  - JOUR
AU  - Campbell, C. M.
AU  - Miyamoto, I.
AU  - Robertson, E. F.
AU  - Williams, P. D.
TI  - The efficiency of PSL(2, p)3 and other direct products of groups
JO  - Glasgow mathematical journal
PY  - 1997
SP  - 259
EP  - 268
VL  - 39
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500032195/
DO  - 10.1017/S0017089500032195
ID  - 10_1017_S0017089500032195
ER  - 
%0 Journal Article
%A Campbell, C. M.
%A Miyamoto, I.
%A Robertson, E. F.
%A Williams, P. D.
%T The efficiency of PSL(2, p)3 and other direct products of groups
%J Glasgow mathematical journal
%D 1997
%P 259-268
%V 39
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500032195/
%R 10.1017/S0017089500032195
%F 10_1017_S0017089500032195

[1] 1.Baik, Y. G. and Pride, S. J., On the efficiency of Coxeter groups, Bull. London Math. Soc. 29 (1997), 32–36. Google Scholar | DOI

[2] 2.Bogley, W. A. and Pride, S. J., Computing generators of π, in Two-dimensional Homotopy and Combinatorial Group theory (Hog-Angeloni, C., Metzler, W. and Sieradski, A. J. eds.), London Math. Soc. Lecture Notes 197 (Cambridge University Press, Cambridge, 1995), 157–188. Google Scholar

[3] 3.Brookes, M. J., Campbell, C. M. and Robertson, E. F., Efficiency and direct products of groups, in Proc. Groups - Korea 1994 (Kim, A. C. and Johnson, D. L. eds.), (Walter de Gruyter, 1995) 25–33. Google Scholar

[4] 4.Campbell, C. M., Kawamata, T., Miyamoto, I., Robertson, E. F. and Williams, P. D., Deficiency zero presentations for certain perfect groups, Proc. Roy. Soc. Edinburgh Sect. A 103 (1986), 63–71. Google Scholar | DOI

[5] 5.Campbell, C. M., Robertson, E. F. and Williams, P. D., On the efficiency of some direct powers of groups, in Groups-Canberra 1989 (Kovács, L. G. ed.), Lecture Notes in Mathematics No. 1456 (Springer-Verlag, Berlin, 1990), 106–113. Google Scholar PubMed | DOI

[6] 6.Campbell, C. M., Robertson, E. F. and Williams, P. D., Efficient presentations of the groups PSL(2, p) × PSL(2, p), p prime, J. London Math. Soc. (2) 41 (1990), 69–77. Google Scholar | DOI

[7] 7.Campbell, C. M., Robertson, E. F. and Williams, P. D., Efficient presentations of direct powers of imperfect groups, Algebra Colloq. 4 (1997), 21–27. Google Scholar

[8] 8.Cannon, J. J., McKay, J. and Young, K. C., The non-abelian simple groups G, |G| < 105—presentations, Comm. Algebra 7 (1979), 1397–1406. Google Scholar | DOI

[9] 9.Conway, J. H., Coxeter, H. S. M. and Shephard, G. C., The centre of a finitely generated group, Tensor 25 (1972), 405–418. Google Scholar

[10] 10.Hert, T. and Williams, P. D., A note on a presentation of PGL (2,p) to appear. Google Scholar

[11] 11.Jamali, A., Computing with simple groups: maximal subgroups and presentations, Ph.D. thesis, University of St Andrews (1988). Google Scholar

[12] 12.Kenne, P. E., Presentations for some direct products of groups, Bull. Austral. Math. Soc. 28 (1983), 131–133. Google Scholar | DOI

[13] 13.Kenne, P. E., Minimal group presentations: a computational approach, Ph.D. Thesis, Australian National University, 1991. Google Scholar

[14] 14.Kovács, L. G., Finite groups with trivial multiplicator and large deficiency, in Proc. Groups – Korea 1994 (Kim, A. C. and Johnson, D. L. eds.), (Walter de Gruyter, 1995), 211–225. Google Scholar

[15] 15.Robertson, E. F., Thomas, R. M. and Wotherspoon, C. I., A class of inefficient groups with symmetric presentations, in Proc. Groups – Korea 1994 (Kim, A. C. and Johnson, D. L. eds.), (Walter de Gruyter, 1995), 277–284. Google Scholar

[16] 16.Sunday, J. G., Presentations of the groups SL(2, m) and PSL(2, m), Canad. J. Math 24 (1972), 1129–1131. Google Scholar | DOI

[17] 17.Suzuki, M., Group theory I, Grundlehren der Mathematischen Wissenschaften 247 (Springer-Verlag, 1982). Google Scholar | DOI

[18] 18.Wiegold, J., The multiplicator of a direct product, Quart. J. Oxford (2) 22 (1971), 103–105. Google Scholar | DOI

[19] 19.Wiegold, J., The Schur multiplier, in Groups – St Andrews 1981, (Campbell, C. M. and Robertson, E. F. eds.), London Mathematical Society Lecture Notes 71 (Cambridge University Press, Cambridge, 1982), 137–154. Google Scholar | DOI

Cité par Sources :