Enumerating word maps in finite groups

Chlebus, Bogdan S.; Cocke, William; Ho, Meng-Che

doi:10.22108/ijgt.2023.136972.1833

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	Enumerating word maps in finite groups
International Journal of Group Theory
مقاله 8، دوره 13، شماره 3، آذر 2024، صفحه 307-318 اصل مقاله (429.45 K)
نوع مقاله: 2022 CCGTA IN SOUTH FLA
شناسه دیجیتال (DOI): 10.22108/ijgt.2023.136972.1833
نویسندگان
Bogdan S. Chlebus¹؛ William Cocke^* ¹؛ Meng-Che Ho²
¹School of Computer and Cyber Sciences, Augusta University, Augusta, GA, USA
²Department of Mathematics, California State University, Northridge, Northridge, CA, USA
چکیده
We consider word maps over finite groups. An $n$-variable word $w$ is an element of the free group on $n$-symbols. For any group $G$, a word $w$ induces a map from $G^n\mapsto G$ where $(g_1,\ldots,g_n)\mapsto w(g_1,\ldots,g_n)$. We observe that many groups have word maps that decompose into components. Such a decomposition facilitates a recursive approach to studying word maps. Building on this observation, and combining it with relevant properties of the word maps, allows us to develop an algorithm to calculate representatives of all the word maps over a finite group. Given these representatives, we can calculate word maps with specific properties over a given group, or show that such maps do not exist. In particular, we have computed an explicit a word on $A_5$ such that only generating tuples are nontrivial in its image. We also discuss how our algorithm could be used to computationally address many open questions about word maps. Promising directions of potential applications include Amit's conjecture, questions of chirality and rationality, and the search for multilinear maps over a group. We conclude with open questions regarding these problems.
کلیدواژه‌ها
Word maps؛ Relatively free groups؛ Algorithms on groups؛ Amit--Ashurst conjecture

مراجع
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Enumerating word maps in finite groups