Elementary Abelian Sylow Subgroups of the Multiplicative Group

Authors

• Geremías Polanco, Smith CollegeFollow

Document Type

Article

Publication Date

4-2026

Publication Title

Journal of Number Theory

Abstract

Erdős and Pomerance have shown that 𝜑⁡(𝑛) typically has about 1 2 ⁢(log⁡log⁡𝑛)2 distinct prime factors. More precisely, 𝜔⁡(𝜑⁡(𝑛)) has normal order 1 2 ⁢(log⁡log⁡𝑛)2. Since 𝜑⁡(𝑛) is the size of the multiplicative group (𝐙/𝑛⁢𝐙)×, this result also gives the normal number of Sylow subgroups of (𝐙/𝑛⁢𝐙)×. Recently, Pollack considered specifically noncyclic Sylow subgroups of (𝐙/𝑛⁢𝐙)×, showing that the count of those has normal order log⁡log⁡𝑛/⁢log⁡log⁡log⁡𝑛. We prove that the count of noncyclic Sylow subgroups that are elementary abelian of a fixed rank 𝑘 ≥2 has normal order 1 𝑘⁢(𝑘−1) ⁢log⁡log⁡𝑛/⁢log⁡log⁡log⁡𝑛. So for example, (typically) among the primes p for which the p-primary component of (𝐙/𝑛⁢𝐙)× is noncyclic, this component is 𝐙/𝑝⁢𝐙 ⊕𝐙/𝑝⁢𝐙 about half the time. Additionally, we show that the count of p for which the p-Sylow subgroup of (𝐙/𝑛⁢𝐙)× is not elementary abelian has normal order 2⁢√𝜋⁢√log⁡log⁡𝑛/log⁡log⁡log⁡𝑛.

Keywords

Multiplicative group; Noncyclic Sylow subgroups; Group of units; Normal order; Elementary abelian group

Volume

281

First Page

205

Last Page

223

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Rights

Licensed to Smith College and distributed CC-BY 4.0 under the Smith College Faculty Open Access Policy.

Version

Author's Accepted Manuscript

Recommended Citation

Polanco, Geremías, "Elementary Abelian Sylow Subgroups of the Multiplicative Group" (2026). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/215