Presentations of finite simple groups: A quantitative approach

Guralnick, Robert M.; Kantor, William M.; Kassabov, Martin; Lubotzky, Alexander

doi:10.1090/s0894-0347-08-00590-0

Cited by 34 publications

(132 citation statements)

References 55 publications

Supporting

Mentioning

130

Contrasting

Order By: Relevance

“…As in the previous case, applying [8, Section 9.4] we obtain that there exists a presentation 1 of G 1 such that l. 1 / Ä C 0 . [8,Section 9.4] in the former case, and using the result of [3] in the latter one, we obtain that G 2 has a 'bounded' presentation 2 with l. 2 / Ä 3C 2 0 . Now, using the results of Tits [11] and Abramenko and Muhlherr [1], we observe that not only G 0 , G 1 and G 2 generate G (as for every pair x, y from a finite set of generators of G there exists an i 2 ¹0; 1; 2º such that x; y 2 G i ), but in fact G is their completed amalgam.…”

Section: Resultsmentioning

confidence: 62%

“…Then G 0 is a quasisimple group of Lie type and by a generalization of a result of Guralnick et al [8,Section 9.4], there exists a constant C 0 and a presentation 0 of G 0 with l. 0 / Ä C 0 . Now, consider 1 D ¹˛nº.…”

Section: Resultsmentioning

confidence: 92%

“…A natural question to consider would be whether the result of Guralnick et al [8] holds in some form for Kac-Moody groups, the infinite-dimensional analogues of finite groups of Lie type.…”

Section: Presentations Of Kac-moody Groupsmentioning

confidence: 99%

“…[7, p. 79]) allowed Korchagina and Lubotzky [10] to produce presentations of untwisted groups of Lie type with l. / D O.n 2 / where n denotes the Lie rank of a corresponding group. Finally, the question was settled once and for all (with the possible exception of the Ree groups 2 G 2 .q/) in the paper of Guralnick, Kantor, Kassabov and Lubotzky [8], where they proved the following result.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Bounded presentations of Kac–Moody groups

Capdeboscq¹

2013

Journal of Group Theory

View full text Add to dashboard Cite

Abstract. In this article we extend the result of Guralnick, Kantor, Kassabov and Lubotzky to the affine Kac-Moody groups: we show that there exists a constant C > 0 such that every affine Kac-Moody group defined over a finite field F q , q 4 (with the exception of e A 1 and e A 1 ), has a presentation with l. / < C . We then derive the consequences of this result for the 2-spherical Kac-Moody groups.

show abstract

Section: Resultsmentioning

confidence: 62%

Section: Resultsmentioning

confidence: 92%

“…A natural question to consider would be whether the result of Guralnick et al [8] holds in some form for Kac-Moody groups, the infinite-dimensional analogues of finite groups of Lie type.…”

Section: Presentations Of Kac-moody Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bounded presentations of Kac–Moody groups

Capdeboscq¹

2013

Journal of Group Theory

View full text Add to dashboard Cite

show abstract

“…For most classes of finite simple groups, Guralnick et al [8] obtained a presentation for each member G that is very short compared to |G|.…”

Section: Proof (I) Let ψ Bementioning

confidence: 99%

Describing finite groups by short first-order sentences

Nies

Tent

2017

Isr. J. Math.

View full text Add to dashboard Cite

Abstract. We say that a class of finite structures for a finite firstorder signature is r-compressible for an unbounded function r : N → N + if each structure G in the class has a first-order description of size at most O(r(|G|)). We show that the class of finite simple groups is logcompressible, and the class of all finite groups is log 3 -compressible. As a corollary we obtain that the class of all finite transitive permutation groups is log 3 -compressible. The results rely on the classification of finite simple groups, the bi-interpretability of the twisted Ree groups with finite difference fields, the existence of profinite presentations with few relators for finite groups, and group cohomology. We also indicate why the results are close to optimal.

show abstract

Towards Constructing Fully Homomorphic Encryption without Ciphertext Noise from Group Theory

Nuida

2020

International Symposium on Mathematics, Quantum Theory, and Cryptography

View full text Add to dashboard Cite

In CRYPTO 2008, 1 year earlier than Gentry’s pioneering “bootstrapping” technique for the first fully homomorphic encryption (FHE) scheme, Ostrovsky and Skeith III had suggested a completely different approach towards achieving FHE. They showed that the $$\mathsf {NAND}$$ operator can be realized in some non-commutative groups; consequently, homomorphically encrypting the elements of the group will yield an FHE scheme, without ciphertext noise to be bootstrapped. However, no observations on how to homomorphically encrypt the group elements were presented in their paper, and there have been no follow-up studies in the literature. The aim of this paper is to exhibit more clearly what is sufficient and what seems to be effective for constructing FHE schemes based on their approach. First, we prove that it is sufficient to find a surjective homomorphism $$\pi :\widetilde{G} \rightarrow G$$ between finite groups for which bit operators are realized in G and the elements of the kernel of $$\pi $$ are indistinguishable from the general elements of $$\widetilde{G}$$. Secondly, we propose new methodologies to realize bit operators in some groups G. Thirdly, we give an observation that a naive approach using matrix groups would never yield secure FHE due to an attack utilizing the “linearity” of the construction. Then we propose an idea to avoid such “linearity” by using combinatorial group theory. Concretely realizing FHE schemes based on our proposed framework is left as a future research topic.

show abstract

Presentations of finite simple groups: A quantitative approach

Abstract: There is a constant C 0 C_0 such that all nonabelian finite simple groups of rank n n over F q \mathbb {F}_q , with the possible exception of the Ree groups 2 G 2 ( 3 2 e + 1 ) ^2G_2(3^{2e+1}) , have presentations with at most … Show more

Cited by 34 publications

References 55 publications

Bounded presentations of Kac–Moody groups

Bounded presentations of Kac–Moody groups

Describing finite groups by short first-order sentences

Towards Constructing Fully Homomorphic Encryption without Ciphertext Noise from Group Theory

Contact Info

Product

Resources

About