Automorphic word maps and the Amit–Ashurst conjecture
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Harish Kishnani
Abstract
In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel) 115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of [A. Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, J. Algebra 485 (2017), 230–246] using elementary techniques.
Acknowledgements
The second named author acknowledges the support from Prime Minister Research Fellowship. We are thankful to William Cocke whose survey talk introduced us to Amit conjecture, and to Josu Sangroniz for e-mail correspondence. We are grateful to the referee for carefully reading this work and providing invaluable suggestions to correct an earlier version of Corollary 3.2.
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Communicated by: Timothy C. Burness
References
[1] A. Amit and U. Vishne, Characters and solutions to equations in finite groups, J. Algebra Appl. 10 (2011), no. 4, 675–686. 10.1142/S0219498811004690Search in Google Scholar
[2] C. Ashurst, Fibres of words in finite groups, a probabilistic approach, PhD thesis, University of Bath, 2012. Search in Google Scholar
[3] R. D. Camina, W. L. Cocke and A. Thillaisundaram, The Amit–Ashurst conjecture for finite metacyclic 𝑝-groups, Eur. J. Math. 9 (2023), no. 3, Paper No. 46. 10.1007/s40879-023-00644-xSearch in Google Scholar
[4] R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel) 115 (2020), no. 6, 599–609. 10.1007/s00013-020-01504-wSearch in Google Scholar
[5] W. Cocke, Two characterizations of finite nilpotent groups, J. Group Theory 21 (2018), no. 6, 1111–1116. 10.1515/jgth-2018-0029Search in Google Scholar
[6] W. Cocke and M.-C. Ho, On the symmetry of images of word maps in groups, Comm. Algebra 46 (2018), no. 2, 756–763. 10.1080/00927872.2017.1327065Search in Google Scholar
[7] W. Cocke and M.-C. Ho, The probability distribution of word maps on finite groups, J. Algebra 518 (2019), 440–452. 10.1016/j.jalgebra.2018.10.022Search in Google Scholar
[8] R. Cortini, On special 𝑝-groups, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 3, 677–689. Search in Google Scholar
[9] A. Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, J. Algebra 485 (2017), 230–246. 10.1016/j.jalgebra.2017.05.017Search in Google Scholar
[10] D. Kaur, H. Kishnani and A. Kulshrestha, Word images and their impostors in finite nilpotent groups, preprint (2022), https://arxiv.org/abs/2205.15369. Search in Google Scholar
[11] M. Levy, On the probability of satisfying a word in nilpotent groups of class 2, preprint (2011), https://arxiv.org/abs/1101.4286. Search in Google Scholar
[12] J. Nielsen, Die Isomorphismengruppe der freien Gruppen, Math. Ann. 91 (1924), no. 3–4, 169–209. 10.1007/BF01556078Search in Google Scholar
[13] D. Puder and O. Parzanchevski, Measure preserving words are primitive, J. Amer. Math. Soc. 28 (2015), no. 1, 63–97. 10.1090/S0894-0347-2014-00796-7Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- The relational complexity of linear groups acting on subspaces
- Cliques in derangement graphs for innately transitive groups
- Representation zeta function of a family of maximal class groups: Non-exceptional primes
- Character degrees of 5-groups of maximal class
- Automorphic word maps and the Amit–Ashurst conjecture
- Groups with subnormal or modular Schmidt 𝑝𝑑-subgroups
- Finite normal subgroups of strongly verbally closed groups
- The central tree property and algorithmic problems on subgroups of free groups
- Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups
- Isomorphisms and commensurability of surface Houghton groups
Articles in the same Issue
- Frontmatter
- The relational complexity of linear groups acting on subspaces
- Cliques in derangement graphs for innately transitive groups
- Representation zeta function of a family of maximal class groups: Non-exceptional primes
- Character degrees of 5-groups of maximal class
- Automorphic word maps and the Amit–Ashurst conjecture
- Groups with subnormal or modular Schmidt 𝑝𝑑-subgroups
- Finite normal subgroups of strongly verbally closed groups
- The central tree property and algorithmic problems on subgroups of free groups
- Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups
- Isomorphisms and commensurability of surface Houghton groups
