The geometry and cohomology of the Mathieu group M12

Ádem, Alejandro; Maginnis, John; Milgram, R. James

doi:10.1016/0021-8693(91)90285-g

Cited by 15 publications

(11 citation statements)

References 13 publications

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“…(The situation for (1.1 (2)) is identical to that for and is Cohen±Macaulay over the Dickson algebra [4]. In particular, since the multiple image classes in H Ã EY F 2 modulo the classes which live only on this summand are F 2 d 8 Y d 12 1Y w 3 from [5], it follows easily that the only possible multiple image classes among the A 7 invariants are…”

Section: The Main Resultsmentioning

confidence: 92%

“…There are three conjugacy classes of extremal 2-elementary subgroups in M 12 , and with respect to the usual embedding into S 12 , given for example in [ and so M D3 V 2 Â V 3 r S 12 . Additionally, in [2] it was shown that restriction to these three subgroups detects H Ã M 12 Y F 2 . Since H Ã S n Y F 2 is also detected by restriction to its elementary 2-subgroups for all n, we need only study restriction to these three groups to determine entirely the image of restriction to H Ã M 12 Y F 2 .…”

Section: X2mentioning

confidence: 97%

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The cohomology of the Mathieu group M23

Milgram¹

2000

Journal of Group Theory

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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 conjecture was orginally made by Loday in the mid-1970s and it is based on Quillen's calculations of the cohomology of many of the families of groups of Lie type a few years earlier; see Gi¨en [10]. The ®rst Mathieu group M 11 and the ®rst Janko group J 1 also satisfy OutG MultG 1, but for both of these groups H 3 GY Z H 0. It would be tempting to amend the conjecture. It is very likely that it only fails for a very small number of the sporadic groups among the simple groups. So one might well suspect that there is a (small) ®nite number n so that H i GY Z 0 for 0`i n implies that G f1g if G is ®nite. But we have no idea as to a suitable candidate for n. At the prime 2 we have that H 6 M 23 Y Z Za2 is the ®rst non-zero homology group. In particular, when we look at the usual inclusion M 23 r S 23 we can ask about the image of this ®rst non-trivial class. We have the well known equivalence B S y p Q 0 S 0 , the 0-component of the in®nite loop space lim n3y n S n , and at the prime 2, Q 0 S 0 2 p imJ 2 Â CokerJ 2 is the Quillen±Sullivan splitting; see [13]. Then CokerJ 2 is 5-connected and we have Theorem. The composition B M 23 3 B S 23 3 B S y Q 0 S 0

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Section: The Main Resultsmentioning

confidence: 92%

Section: X2mentioning

confidence: 97%

The cohomology of the Mathieu group M23

Milgram¹

2000

Journal of Group Theory

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“…The existence of such a spectral sequence imposes severé restrictions on the possible shape of the cohomology ring of a finite group. We use this technique for example in this paper in order to prove that the 1 Both authors are partly supported by grants from the NSF.…”

Section: Introduction W Xmentioning

confidence: 97%

Cohomology of the Double Cover of the Mathieu Group M12

Benson

Carlson

2000

Journal of Algebra

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In this paper we calculate the mod two cohomology of the double cover of the Mathieu group M . The starting point is the calculation by Adem, Maginnis and 12 Milgram of the mod two cohomology of M . We use a hypercohomology spectral 12 sequence to determine a differential in the Lyndon-Hochschild-Serre spectral sequence of the central extension, and this gets us as far as the E page. Ungrading requires restriction to the Sylow 2-subgroup, and here some computer calculations come to our rescue.

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“…A computer method for the second homology of a permutation group was illustrated on the Mathieu groups M 21 and M 22 in [15]. The mod p cohomology H * (G, F p ) is now known for all Mathieu groups except M 24 [21,1,2,17]. With the help of the Bockstein spectral sequence it is, in principle, possible to obtain integral homology from mod p cohomology (p ranging over the prime divisors of the group order), though the details can be difficult.…”

Section: Introductionmentioning

confidence: 99%

Wythoff polytopes and low-dimensional homology of Mathieu groups

Sikirić

Ellis

2009

Journal of Algebra

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We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as H 5 (M 23 , Z) = Z 7 and H 3 (M 24 , Z) = Z 12 . One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free ZM n -resolution. Both methods apply in principle to arbitrary finite groups.

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The geometry and cohomology of the Mathieu group M12

Cited by 15 publications

References 13 publications

The cohomology of the Mathieu group M23

The cohomology of the Mathieu group M23

Cohomology of the Double Cover of the Mathieu Group M12

Wythoff polytopes and low-dimensional homology of Mathieu groups

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