Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 059, 38 pages      arXiv:1810.00463
Contribution to the Special Issue on Moonshine and String Theory

Third Homology of some Sporadic Finite Groups

Theo Johnson-Freyd a and David Treumann b
a) Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada
b) Department of Mathematics, Boston College, Boston, Massachusetts, USA

Received September 30, 2018, in final form August 06, 2019; Published online August 10, 2019

We compute the integral third homology of most of the sporadic finite simple groups and of their central extensions.

Key words: sporadic groups; group cohomology.

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  1. Adem A., Milgram R.J., The subgroup structure and mod $2$ cohomology of O'Nan's sporadic simple group, J. Algebra 176 (1995), 288-315.
  2. Adem A., Milgram R.J., Cohomology of finite groups, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Vol. 309, Springer-Verlag, Berlin, 2004.
  3. Cartan H., Eilenberg S., Homological algebra, Princeton University Press, Princeton, N. J., 1956.
  4. Cheng M.C.N., de Lange P., Whalen D.P.Z., Generalised umbral moonshine, SIGMA 15 (2019), 014, 27 pages, arXiv:1608.07835.
  5. Cohen A.M., Wales D.B., Finite subgroups of $F_4({\bf C})$ and $E_6({\bf C})$, Proc. London Math. Soc. 74 (1997), 105-150.
  6. Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A., Atlas of finite groups: maximal subgroups and ordinary characters for simple groups (with computational assistance from J.G. Thackray), Oxford University Press, Eynsham, 1985.
  7. Conway J.H., Norton S.P., Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308-339.
  8. Curtis M., Wiederhold A., Williams B., Normalizers of maximal tori, in Localization in Group Theory and Homotopy Theory, and Related Topics (Sympos., Battelle Seattle Res. Center, Seattle, Wash., 1974), Lecture Notes in Math., Vol. 418, Springer, Berlin, 1974, 31-47.
  9. Dempwolff U., On extensions of an elementary abelian group of order $2^{5}$ by ${\rm GL}(5,2)$, Rend. Sem. Mat. Univ. Padova 48 (1972), 359-364.
  10. Dempwolff U., On the second cohomology of ${\rm GL}(n,2)$, J. Austral. Math. Soc. 16 (1973), 207-209.
  11. Duncan J.F., Super-moonshine for Conway's largest sporadic group, Duke Math. J. 139 (2007), 255-315, arXiv:math.RT/0502267.
  12. Duncan J.F.R., Vertex operators, and three sporadic groups, Ph.D. Thesis, Yale University, 2006.
  13. Duncan J.F.R., Mertens M.H., Ono K., O'Nan moonshine and arithmetic, arXiv:1702.03516.
  14. Dutour Sikirić M., Ellis G., Wythoff polytopes and low-dimensional homology of Mathieu groups, J. Algebra 322 (2009), 4143-4150, arXiv:0812.4291.
  15. Frame J.S., The characters of the Weyl group $E_{8}$, in Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon, Oxford, 1970, 111-130.
  16. Gaberdiel M.R., Persson D., Ronellenfitsch H., Volpato R., Generalized Mathieu Moonshine, Commun. Number Theory Phys. 7 (2013), 145-223, arXiv:1211.7074.
  17. Gannon T., Much ado about Mathieu, Adv. Math. 301 (2016), 322-358, arXiv:1211.5531.
  18. Ganter N., Hecke operators in equivariant elliptic cohomology and generalized Moonshine, in Groups and Symmetries, CRM Proc. Lecture Notes, Vol. 47, Amer. Math. Soc., Providence, RI, 2009, 173-209, arXiv:0706.2898.
  19. Green D.J., On the cohomology of the sporadic simple group $J_4$, Math. Proc. Cambridge Philos. Soc. 113 (1993), 253-266.
  20. Griess Jr R.L., Schur multipliers of the known finite simple groups, Ph.D. Thesis, The University of Chicago, 1971.
  21. Griess Jr. R.L., Automorphisms of extra special groups and nonvanishing degree $2$ cohomology, Pacific J. Math. 48 (1973), 403-422.
  22. Griess Jr. R.L., On a subgroup of order $2^{15}\mid GL(5,2)\mid $ in $E_{8}({\bf C}),$ the Dempwolff group and ${\rm Aut}(D_{8}\circ D_{8}\circ D_{8})$, J. Algebra 40 (1976), 271-279.
  23. Grodal J., Low dimensional homology of finite groups of Lie type, away from the characteristic, unpublished.
  24. Henriques A., The classification of chiral WZW models by $H^4_+(BG,{\mathbb Z})$, in Lie Algebras, Vertex Operator Algebras, and Related Topics,Contemp. Math., Vol. 695, Amer. Math. Soc., Providence, RI, 2017, 99-121, arXiv:1602.02968.
  25. Hochschild G., Serre J.-P., Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953), 110-134.
  26. Ivanov A.A., The Monster group and Majorana involutions, Cambridge Tracts in Mathematics, Vol. 176, Cambridge University Press, Cambridge, 2009.
  27. Johnson-Freyd T., The moonshine anomaly, Comm. Math. Phys. 365 (2019), 943-970, arXiv:1707.08388.
  28. Johnson-Freyd T., Treumann D., $\mathrm{H}^4(\mathrm{Co}_0;\mathbf{Z}) = \mathbf{Z}/24$, Int. Math. Res. Not., to appear, arXiv:1707.07587.
  29. Lyons R., Evidence for a new finite simple group, J. Algebra 20 (1972), 540-569.
  30. Mason G., Reed-Muller codes, the fourth cohomology group of a finite group, and the $\beta$-invariant, J. Algebra 312 (2007), 218-227.
  31. Milgram R.J., On the geometry and cohomology of the simple groups $G_2(q)$ and ${}_3D_4(q)$. II, in Group Representations: Cohomology, Group Actions and Topology (Seattle, WA, 1996), Proc. Sympos. Pure Math., Vol. 63, Amer. Math. Soc., Providence, RI, 1998, 397-418.
  32. Milgram R.J., The cohomology of the Mathieu group $M_{23}$, J. Group Theory 3 (2000), 7-26.
  33. Serre J.-P., Homologie singulière des espaces fibrés. Applications, Ann. of Math. 54 (1951), 425-505.
  34. Smith P.E., A simple subgroup of $M?$ and $E_{8}(3)$, Bull. London Math. Soc. 8 (1976), 161-165.
  35. Soicher L.H., Presentations for Conway's group ${\rm Co}_1$, Math. Proc. Cambridge Philos. Soc. 102 (1987), 1-3.
  36. Thomas C.B., Characteristic classes and the cohomology of finite groups, Cambridge Studies in Advanced Mathematics, Vol. 9, Cambridge University Press, Cambridge, 1987.
  37. Thomas C.B., Characteristic classes and $2$-modular representations of some sporadic simple groups, in Algebraic Topology (Evanston, IL, 1988), Contemp. Math., Vol. 96, Amer. Math. Soc., Providence, RI, 1989, 303-318.
  38. Thomas C.B., Characteristic classes and $2$-modular representations for some sporadic simple groups. II, in Algebraic Topology Poznań 1989, Lecture Notes in Math., Vol. 1474, Springer, Berlin, 1991, 371-381.
  39. Thompson J.G., A conjugacy theorem for $E_{8}$, J. Algebra 38 (1976), 525-530.
  40. Tits J., Normalisateurs de tores. I. Groupes de Coxeter étendus, J. Algebra 4 (1966), 96-116.
  41. Weibel C.A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.
  42. Wilson R., Walsh P., Tripp J., Suleiman I., Parker R., Norton S., Nickerson S., Linton S., Bray J., Abbott R., ATLAS of finite group representations, Version 3, available at
  43. Wilson R.A., The complex Leech lattice and maximal subgroups of the Suzuki group, J. Algebra 84 (1983), 151-188.
  44. Wilson R.A., Maximal subgroups of sporadic groups, in Finite Simple Groups: Thirty Years of the Atlas and Beyond,Contemp. Math., Vol. 694, Amer. Math. Soc., Providence, RI, 2017, 57-72, arXiv:1701.02095.

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