We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime q and a finite abelian q-group H, we consider the set of integers n ≤ x such that the Sylow q-subgroup of the multiplicative group (ℤ/nℤ)× is isomorphic to H. We show that the counting function of this set of integers is asymptotic to Kx(loglog x)ℓ/(log x)1/(q−1) for explicit constants K and ℓ depending on q and H. Second, we consider the set of integers n ≤ x such that the multiplicative group (ℤ/nℤ)× is “maximally non-cyclic”, that is, such that all of its prime-power subgroups are elementary groups. We show that the counting function of this set of integers is asymptotic to Ax/(log x)1−ξ for an explicit constant A, where ξ is Artin’s constant. As it turns out, both of these group-theoretic problems can be reduced to problems of counting integers with restrictions on their prime factors, allowing them to be addressed by classical techniques of analytic number theory.

中文翻译：

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计算具有规定子组的乘法组

我们研究了两个计数问题，这些问题表面上看起来非常具有群论性，但经过仔细研究，结果却发现它们涉及对质因数有限制的整数。首先，给定一个奇数素数q和有限阿贝尔q-团体H，我们考虑整数集n ≤ X使得 Sylowq-乘法群的子群(ℤ/nℤ)×同构于H. 我们证明了这组整数的计数函数是渐近的ķX(日志日志 X)ℓ/(日志 X)1/(q-1)对于显式常量ķ和ℓ根据q和H. 其次，我们考虑整数集n ≤ X使得乘法群(ℤ/nℤ)×是“最大非循环的”，也就是说，它的所有素数次幂子群都是基本群。我们证明了这组整数的计数函数是渐近的一种X/(日志 X)1-ξ对于显式常数一种， 在哪里ξ是阿廷常数。事实证明，这两个群论问题都可以简化为计算整数的问题，但它们的素因数受到限制，从而可以通过解析数论的经典技术来解决。