Exponents of modules and maps (Q1118038)

In an earlier paper [Proc. Am. Math. Soc. 102, 814-816 (1988; Zbl 0645.20032)] the author established a surprising connection between the structure of a group G and the exponent of its integral cohomology; the precise nature of this connection however remained obscure. Here this result is extended and made more precise. It is shown that the homogeneous elements \(\zeta_ 1,\zeta_ 2,...,\zeta_ r\in H^*(G,{\mathbb{Z}})\) have the property that \(| G|\) divides \(\prod^{r}_{i=1}\exp (\zeta_ i)\), provided the set consisting of \(\zeta_ 1,\zeta_ 2,...,\zeta_ r\) satisfies a certain condition concerned with the maximal ideal spectrum of the ring \(H^*(G,{\mathbb{Z}})\). (The condition implies in particular that \(r\geq p\)-rank of G for all primes p dividing \(| G|.)\) An example of a group G of order \(p^ 5\) shows that, in general, \(| G|\) is a proper divisor of \(\prod^{r}_{i=1}\exp (\zeta_ i)\), for all choices \(\zeta_ 1,\zeta_ 2,...,\zeta_ r\in H^*(G,{\mathbb{Z}})\) satisfying this condition. As an application the author deduces the result that for an extra special p-group of order \(p^{2n+1}\) (p odd) any element \(\zeta \in H^{2m}(G,{\mathbb{Z}})\), \(m\geq 1\), whose restriction to the Frattini subgroup of G is non-trivial, has exponent \(p^{n+1}\). - The result on the integral cohomology stated above, is obtained as a special case of a more general theorem on the cohomology of an RG-lattice M, where R is a ``nice'' principal ideal domain. It relates the exponent of the cohomology of M to the ``exponent of M''; the author defines the exponent of an RG-lattice M as the order of the identity \(1_ M: M\to M\) modulo the endomorphisms that factor through a projective RG-lattice.