2011/12/07 by Jonathan Pakianathan, Pakianathan, Jonathan
Mathematics · #Algebraic structures and combinatorial models #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology #math.AT #math.GR #math.KT #msc:20J05 #msc:20J06
paper · pdf · doi:10.48550/arxiv.1112.1671
5 pages. Withdrawn due to significant overlap with prior work of Kuo, Tzee-nan till later determination if differences are significant enough to publish
arxiv created 2011/12/08 · arxiv updated 2011/12/09
It is well known that the positive degree cohomology of a finite group G is annihilated by |G|. We improve on this bound in the case of odd degree elements in the integer cohomology ring and show that eodd(G), the exponent of the ⊕k=0∞ H2k+1(G,ℤ) satisfies eodd(G)2 divides 2|G| and in particular eodd(G) ≤ √(2|G|). We also provide examples to show this bound for eodd(G) is sharp as a general bound over all finite groups G. The result comes from a fact about zero divisors having "complementary exponent" which we prove using duality in Tate cohomology. More particularly if α, β are elements of positive degree in H^*(G,ℤ) satisfying αβ= 0 then the order of β, o(β) divides (|G|)/(o(α)). We also apply this fact to get some results on elements of exceptionally high exponent in the cohomology ring.