Journal
JOURNAL OF ALGEBRA
Volume 569, Issue -, Pages 195-219Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jalgebra.2020.10.031
Keywords
Cayley graph; Babai conjecture; Special linear group; Transvection
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Funding
- European Research Council (ERC) under the European Union [741420]
- National Research Development and Innovation Office (NKFIH) [K115799]
- Janos Bolyai Research Scholarship of the Hungarian Academy of Sciences
- European Research Council (ERC) [741420] Funding Source: European Research Council (ERC)
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The paper proves that for SL(n, p) with a generating set X containing a transvection, the diameter of Cay(G, X) is bounded by (log |G|) for some absolute constant. A similar result is also shown for G = SL(n, K), where K can be any field.
A well-known conjecture of Babai states that if G is a finite simple group and X is a generating set of G, then the diameter of the Cayley graph Cay(G, X) is bounded above by (log vertical bar G vertical bar)(c) for some absolute constant c. The goal of this paper is to prove such a bound for the diameter of Cay(G, X) whenever G = SL(n, p) and X is a generating set of G which contains a transvection. A natural analogue of this result is also proved for G = SL(n, K), where K can be any field. (C) 2020 The Author. Published by Elsevier Inc.
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