TY - JOUR
T1 - ENUMERATING WORD MAPS IN FINITE GROUPS
AU - Chlebus, Bogdan S.
AU - Cocke, William
AU - Ho, Meng Che
N1 - Publisher Copyright:
© 2024 University of Isfahan.
PY - 2024/9
Y1 - 2024/9
N2 - 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 Gn ↦→ G where (g1, …, gn) ↦→ w(g1, …, gn). 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 A5 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.
AB - 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 Gn ↦→ G where (g1, …, gn) ↦→ w(g1, …, gn). 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 A5 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.
KW - Algorithms on groups
KW - Amit–Ashurst conjecture
KW - Relatively free groups
KW - Word maps
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U2 - 10.22108/ijgt.2023.136972.1833
DO - 10.22108/ijgt.2023.136972.1833
M3 - Article
AN - SCOPUS:85181696242
SN - 2251-7650
VL - 13
SP - 307
EP - 318
JO - International Journal of Group Theory
JF - International Journal of Group Theory
IS - 3
ER -