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On the cohomology of the sporadic simple group J4

Published online by Cambridge University Press:  24 October 2008

David John Green
David John Green
Affiliation:
Institut für Experimentelle Mathematik, Universität GHS Essen, D-W-4300 Essen 12, Germany
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Extract

In this paper we calculate part of the integral cohomology ring of the sporadic simple group J4; this group has order 221.33.5.7. 113.23.29.31.37.43. More precisely, we obtain all of the cohomology ring except for the 2-primary part. As the cohomology has already been written down [9] at the primes which divide the group order only once, we concentrate here on the primes 3 and 11. In both of these cases the Sylow p-subgroups are extraspecial of order p3 and exponent p. We use the method which identifies the p-primary cohomology with the ring of stable classes in the cohomology of a Sylow p-subgroup. The stable classes are all invariant under the action of the Sylow p-normalizer; and some time is spent finding invariant classes in the cohomology ring of , the extraspecial group. Section 2 studies the prime 11: the invariant classes are the stable classes, because the Sylow 11-subgroups have the Trivial Intersection (T.I.) property. In Section 3 we study the prime 3, and see that all conditions for invariant classes to be stable reduce to one condition.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 2 , March 1993 , pp. 253 - 266
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

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