• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.815-827
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• 2019

It is well known that the denominator of the Dedekind sum s(c, d) divides 2 gcd(d, 3)d and that no smaller denominator independent of c can be expected. In contrast, here we prove that we usually get a smaller denominator in S(H, d), the sum of the s(c, d)'s over all the c's in a subgroup H of order n > 1 in the multiplicative group $(\mathbb{Z}/d\mathbb{Z})^*$. First, we prove that for p > 3 a prime, the sum 2S(H, p) is a rational integer of the same parity as (p-1)/2. We give an application of this result to upper bounds on relative class numbers of imaginary abelian number fields of prime conductor. Finally, we give a general result on the denominator of S(H, d) for non necessarily prime d's. We show that its denominator is a divisor of some explicit divisor of 2d gcd(d, 3).

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1835-1843
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• 2018

In this paper, finite p-groups G satisfying $N_G(H){\leq}H^G$ for every non-normal subgroup H of G are completely classified. This solves a problem proposed by Y. Berkovich.