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Groups with many roots | ||
| International Journal of Group Theory | ||
| مقاله 4، دوره 9، شماره 4، اسفند 2020، صفحه 261-276 اصل مقاله (254.71 K) | ||
| نوع مقاله: Proceedings of the conference "Engel conditions in groups" - Bath - UK - 2019 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2020.119870.1582 | ||
| نویسندگان | ||
| Sarah Hart* 1؛ Daniel McVeagh2 | ||
| 1Birbeck, University of London | ||
| 2Department of Economics, Mathematics and Statistics, Birkbeck, University of London | ||
| چکیده | ||
| Given a prime $p$, a finite group $G$ and a non-identity element $g$, what is the largest number of $p^{th}$ roots $g$ can have? We write $ϱ_p(G)$, or just $ϱ_p$, for the maximum value of $\frac{1}{|G|}|\{x \in G: x^p=g\}|$, where $g$ ranges over the non-identity elements of $G$. This paper studies groups for which $ϱ_p$ is large. If there is an element $g$ of $G$ with more $p^{th}$ roots than the identity, then we show $ϱ_p(G) \leq ϱ_p(P)$, where $P$ is any Sylow $p$-subgroup of $G$, meaning that we can often reduce to the case where $G$ is a $p$-group. We show that if $G$ is a regular $p$-group, then $ϱ_p(G) \leq \frac{1}{p}$, while if $G$ is a $p$-group of maximal class, then $ϱ_p(G) \leq \frac{1}{p} + \frac{1}{p^2}$ (both these bounds are sharp). We classify the groups with high values of $ϱ_2$, and give partial results on groups with high values of $ϱ_3$. | ||
| کلیدواژه‌ها | ||
| $pth$ roots؛ square roots؛ cube roots | ||
| مراجع | ||
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[10] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language. Computational algebra and number theory (London, 1993), J. Symbolic Comput., 24 (1997) 235–265. [11] C. T. C. Wall, On groups consisting mostly of involutions, Proc. Camb. Phil. Soc., 67 (1970) 251–262. | ||
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