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Title: Elementary abelian Sylow subgroups of the multiplicative group
Erd\H{o}s and Pomerance have shown that $$\varphi(n)$$ typically has about $$\frac{1}{2}(\log\log{n})^2$$ distinct prime factors. More precisely, $$\omega(\varphi(n))$$ has normal order $$\frac{1}{2}(\log\log{n})^2$$. Since $$\varphi(n)$$ is the size of the multiplicative group $$(\Z/n\Z)^{\times}$$, this result also gives the normal number of Sylow subgroups of $$(\Z/n\Z)^{\times}$$. Recently, Pollack considered specifically noncyclic Sylow subgroups of $$(\Z/n\Z)^{\times}$$, 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 $$k\ge 2$$ has normal order $$\frac{1}{k(k-1)} \log\log{n}/\log\log\log{n}$$. So for example, (typically) among the primes $$p$$ for which the $$p$$-primary component of $$(\Z/n\Z)^{\times}$$ is noncyclic, this component is $$\Z/p\Z \oplus \Z/p\Z$$ about half the time. Additionally, we show that the count of $$p$$ for which the $$p$$-Sylow subgroup of $$(\Z/n\Z)^{\times}$$ is not elementary abelian has normal order $$2\sqrt{\pi} \sqrt{\log\log{n}}/\log\log\log{n}$$.  more » « less
Award ID(s):
2316986
PAR ID:
10649934
Author(s) / Creator(s):
; ;
Publisher / Repository:
Elsevier
Date Published:
Journal Name:
Journal of Number Theory
Volume:
281
Issue:
C
ISSN:
0022-314X
Page Range / eLocation ID:
205 to 223
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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