Wythoff polytopes and low-dimensional homology of Mathieu groups

Details Wythoff polytopes and low-dimensional homology of Mathieu groups Wythoff polytopes and low-dimensional homology of Mathieu groups

2008-12-22

arxiv.org/abs/0812.4291v2

Mathematics

Group Theory

10 pages, 1 figure, 4 tables

Scientific paper

We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as $H_5(M_{23},\ZZ)=\ZZ_7$ and $H_3(M_{24},\ZZ)=\ZZ_{12}$. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free $\ZZ M_n$-resolution. Both methods apply in principle to arbitrary finite groups.

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