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2020•Journal of AlgebraOpen access

Diameter of Cayley graphs of SL(n,p) with generating sets containing a transvection

Zoltán Halasi

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Abstract

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⁡|G|)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.

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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⁡|G|)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.

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Available abstract

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⁡|G|)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.

Key concepts: Cayley graph, Generating set of a group, Combinatorics, Mathematics, Conjecture, Bounded function, Graph, Cayley's theorem

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