A diameter bound for finite simple groups of large rank

Biswas, Arindam; Yang, Yilong

doi:10.1112/jlms.12018

Cited by 8 publications

(28 citation statements)

References 17 publications

(28 reference statements)

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“…This is a linear algebraic analogue of the degree reduction lemma for permutations by Babai and Seress [, Lemma 3]. It improves the corresponding one by Biswas and Yang [, Lemma 4.4(ii)]. Theorem then follows by combining Lemma with the rest of the Biswas–Yang machinery.…”

Section: Proof Of Theoremmentioning

confidence: 60%

“…Compare Lemma with [, Lemma 4.4(ii)]. The key difference is that Biswas and Yang required the primes to be coprime with

p (q - 1)

, while we do not have such a restriction.…”

Section: Proof Of Theoremmentioning

confidence: 98%

“…We first state our degree reduction lemma, and indicate how Theorem follows from this together with . We then prove this lemma in Section 2.3.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 2 more Smart Citations

An improved diameter bound for finite simple groups of Lie type

Halasi

Maróti

Pyber

et al. 2019

Bull. London Math. Soc.

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For a finite group G, let diam(G) denote the maximum diameter of a connected Cayley graph of G. A well-known conjecture of Babai states that diam(G) is bounded by (log 2 |G|) O(1) in case G is a non-abelian finite simple group. Let G be a finite simple group of Lie type of Lie rank n over the field Fq. Babai's conjecture has been verified in case n is bounded, but it is wide open in case n is unbounded. Recently, Biswas and Yang proved that diam(G) is bounded by q O(n(log 2 n+log 2 q) 3 ) . We show that in fact diam(G) < q O(n(log 2 n) 2 ) holds. Note that our bound is significantly smaller than the order of G for n large, even if q is large. As an application, we show that more generally diam(H) < q O(n(log 2 n) 2 ) holds for any subgroup H of GL(V ), where V is a vector space of dimension n defined over the field Fq.

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Section: Proof Of Theoremmentioning

confidence: 60%

“…Compare Lemma with [, Lemma 4.4(ii)]. The key difference is that Biswas and Yang required the primes to be coprime with

p (q - 1)

, while we do not have such a restriction.…”

Section: Proof Of Theoremmentioning

confidence: 98%

See 1 more Smart Citation

An improved diameter bound for finite simple groups of Lie type

Halasi

Maróti

Pyber

et al. 2019

Bull. London Math. Soc.

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show abstract

“…The proof of part (iii) relies on [, Theorem 1.4], showing that, for

G = C l_{n} false(q false)

, the diameter

d

of any connected Cayley graph of

G

satisfies

d ⩽ q^{O (n {(log n + log q)}^{3})} .

Combining this with the fact that the girth

g

Γ (G, S)

satisfies

g ⩾ B (k) n

almost surely (see the remark following Theorem ), we easily derive part (iii).…”

Section: Proof Of Propositionmentioning

confidence: 90%

“…The diameter of Cayley graphs of finite simple groups (with explicit or with random generators) has also attracted considerable attention (see, for instance, and the references therein). Clearly if

d

and

g

are the diameter and girth, respectively, then a trivial lower bound for

d

⌊ \frac{g}{2} ⌋

, and there is interest in finding families of graphs for which

d

is bounded in terms of

g

.…”

Section: Introductionmentioning

confidence: 99%

Girth, words and diameter

Liebeck

Shalev

2019

Bull. London Math. Soc.

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We study the girth of Cayley graphs of finite classical groups G on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word w takes the value 1 when evaluated in G in terms of the length of w, which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.

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Babai’s conjecture for high-rank classical groups with random generators

Eberhard¹,

Jezernik

2021

Invent. math.

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Let $$G = {\text {SCl}}_n(q)$$ G = SCl n ( q ) be a quasisimple classical group with n large, and let $$x_1, \ldots , x_k \in G$$ x 1 , … , x k ∈ G be random, where $$k \ge q^C$$ k ≥ q C . We show that the diameter of the resulting Cayley graph is bounded by $$q^2 n^{O(1)}$$ q 2 n O ( 1 ) with probability $$1 - o(1)$$ 1 - o ( 1 ) . In the particular case $$G = {\text {SL}}_n(p)$$ G = SL n ( p ) with p a prime of bounded size, we show that the same holds for $$k = 3$$ k = 3 .

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A diameter bound for finite simple groups of large rank

Cited by 8 publications

References 17 publications

An improved diameter bound for finite simple groups of Lie type

An improved diameter bound for finite simple groups of Lie type

Girth, words and diameter

Babai’s conjecture for high-rank classical groups with random generators

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