Enumerating word maps in finite groups
عنوان مقاله: Enumerating word maps in finite groups
شناسه ملی مقاله: JR_THEGR-13-3_007
منتشر شده در در سال 1403
شناسه ملی مقاله: JR_THEGR-13-3_007
منتشر شده در در سال 1403
مشخصات نویسندگان مقاله:
Bogdan Chlebus - School of Computer and Cyber Sciences, Augusta University, Augusta, GA, USA
William Cocke - School of Computer and Cyber Sciences, Augusta University, Augusta, GA, USA
Meng-Che Ho - Department of Mathematics, California State University, Northridge, Northridge, CA, USA
خلاصه مقاله:
Bogdan Chlebus - School of Computer and Cyber Sciences, Augusta University, Augusta, GA, USA
William Cocke - School of Computer and Cyber Sciences, Augusta University, Augusta, GA, USA
Meng-Che Ho - Department of Mathematics, California State University, Northridge, Northridge, CA, USA
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_۱,\ldots,g_n)\mapsto w(g_۱,\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_۵ 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.
کلمات کلیدی: Word maps, relatively free groups, Algorithms on groups, Amit--Ashurst conjecture
صفحه اختصاصی مقاله و دریافت فایل کامل: https://civilica.com/doc/1947315/