A note on the classgroup of integral group rings of some cyclic groups

Extract

In [1] Fröhlich considers the kernel D(Z(Γ)) of the map of class-groups C(Z(Γ)) → C(), Γ a finite abelian group, the maximal order in the rational group ring Q(Γ). We obtain under mild hypotheses a non-trivial lower bound for the cardinality k(Γ) of the finite group D(Z(Γ)) when Γ is the cyclic group of order 2pⁿ, p an odd prime. In fact, let f be the smallest positive integer such that 2^f ≡ 1 mod.pⁿ, If 2|f then k(Γ) > 1 for pⁿ ≠ 3, 3², 5 and k(Γ) is divisible by primes ≠ 2, p except possibly when pⁿ is a Fermat prime or when pⁿ = 3². The latter result contrasts with the fact that D(Z(Γ)) is a p-group if Γ is a p-group [1; Theorem 5].