2000
DOI: 10.1515/jgth.2000.008
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The cohomology of the Mathieu group M23

Abstract: The Mathieu group M 23 of order 2 7 Á 3 2 Á 5 Á 7 Á 11 Á 23 is one of the 26 sporadic groups. In this note we determine the cohomology rings H Ã M 23 Y F p for all primes p dividing jM 23 j. This has several interesting consequences.Among the sporadic groups M 23 is somewhat unusual in that OutM 23 MultM 23 1Yand, as a result of our calculation H i M 23 Y Z 0 for i`5. In particular M 23 is the ®rst known counter-example to the conjecture that if G is a ®nite group with H i GY Z 0, i 1Y 2Y 3, then G f1g. This c… Show more

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Cited by 6 publications
(7 citation statements)
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“…Indeed, [18] computed that H 3 M 24 (1; U(1)) ∼ = Z 12 , and [31] explicitly verified that a generating cocycle yields the phases appearing in Mathieu Moonshine. Moreover, H 3 M 23 (1; U(1)) ∼ = 1 [41], and so the phases for elements lying in M 23 should be trivial, and this indeed is what is observed. It is also observed that many pairs (g, h) are not viable, as expected since α is nontrivial.…”
Section: Speculationssupporting
confidence: 60%
“…Indeed, [18] computed that H 3 M 24 (1; U(1)) ∼ = Z 12 , and [31] explicitly verified that a generating cocycle yields the phases appearing in Mathieu Moonshine. Moreover, H 3 M 23 (1; U(1)) ∼ = 1 [41], and so the phases for elements lying in M 23 should be trivial, and this indeed is what is observed. It is also observed that many pairs (g, h) are not viable, as expected since α is nontrivial.…”
Section: Speculationssupporting
confidence: 60%
“…For finite groups, for instance, there are well-known constraints due to Maschke (cohomological version quoted in [19, p. 227]), Evens [13], and Swan [39]. In the same vein is Milgram's counterexample in [29] to the conjecture (attributed to Loday) that no nontrivial finite group can have its first three positivedimensional homology groups zero, see [16]. Since any group with a series of finite length whose factors are either infinite cyclic or locally finite has the direct sum of all its reduced homology groups either infinite or zero [9], it is apparent that one needs to focus on a more general class of groups with torsion.…”
Section: Introduction and Statement Of The Main Resultsmentioning
confidence: 99%
“…This will occur for rational homology spheres with a free G-action where H 4 (G, Z) = 0. An example of this phenomenon is given by the Mathieu group M 23 (see [23]).…”
Section: Note That the Stable Map ωmentioning
confidence: 99%