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The pq-condition for -manifold groups

Author(s): Siddhartha Gadgil
Journal: Proc. Amer. Math. Soc. 129 (2001), 1873-1875.
MSC (2000): Primary 57M05, 57M60
Posted: November 30, 2000
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Abstract:

We give an elementary, topological proof of the fact that any subgroup of order of a finite -manifold group is cyclic if and are distinct odd primes. This condition, together with related results of Milnor and Reidemeister, implies that such a group acts orthogonally on some sphere.

References:

1.: E. G. Mennike Finite fundamental groups of three-dimensional manifolds Mat. Zametki 57 (1995), 105-117 MR 97e:57001
2.: J. Milnor Groups which act on without fixed points Jour. Amer. Math. Soc. 79 (1957), 623-630. MR 19:761d
3.: K. Reidemeister Kommutative Fundamentalgrüppen Monatsch. Math. Phy. 43 (1935), 20-28.
4.: P. Smith Permutable periodic transformations Proc. Nat. Acad. Sci. U.S.A. 30 (1944), 105-108. MR 5:274d
5.: H. Zassenhaus Über endliche Fastkörper Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 79 (1936), 187-220.


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