On large subgroups of finite groups

Lev, Arieh

doi:10.1016/0021-8693(92)90041-j

Cited by 7 publications

(8 citation statements)

References 2 publications

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“…These include all simple finite groups other than cyclic groups, the alternating groups, the 26 sporadic groups, and the Tits group. See [Lev92,Wik17] for sources.…”

Section: H| · T (|K|) + |K| · T (|H|) +mentioning

confidence: 99%

A fast generalized DFT for finite groups of Lie type

Hsu¹,

Umans²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We give an arithmetic algorithm using O(|G| ω/2+o(1) ) operations to compute the generalized Discrete Fourier Transform (DFT) over group G for finite groups of Lie type, including the linear, orthogonal, and symplectic families and their variants, as well as all finite simple groups of Lie type. Here ω is the exponent of matrix multiplication, so the exponent ω/2 is optimal if ω = 2.Previously, "exponent one" algorithms were known for supersolvable groups and the symmetric and alternating groups. No exponent one algorithms were known (even under the assumption ω = 2) for families of linear groups of fixed dimension, and indeed the previous bestknown algorithm for SL 2 (F q ) had exponent 4/3 despite being the focus of significant effort. We unconditionally achieve exponent at most 1.19 for this group, and exponent one if ω = 2.We also show that ω = 2 implies a √ 2 exponent for general finite groups G, which beats the longstanding previous best upper bound (assuming ω = 2) of 3/2.

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“…These include all simple finite groups other than cyclic groups, the alternating groups, the 26 sporadic groups, and the Tits group. See [Lev92,Wik17] for sources.…”

Section: H| · T (|K|) + |K| · T (|H|) +mentioning

confidence: 99%

A fast generalized DFT for finite groups of Lie type

Hsu¹,

Umans²

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…As mentioned above, it was proved in [3] that each simple nonabelian group G has a large maximal subgroup. When Out(G) = 1 this large subgroup H certainly satisfies our extra condition.…”

Section: Proof Of Proposition 13mentioning

confidence: 99%

“…Now let l ≥ 2. If l = 2 and q = 3, then either d = 1 and G L 3 (3) or G 2 (3), or d = 2 and G PSp 4 (3). In the first case |B| = 2 2 · 3 3 or 2 2 · 3 6 and |G : B| = 2 2 · 13 or 2 4 · 7 · 13, respectively.…”

Section: Proof Of Proposition 13mentioning

confidence: 99%

“…The case when all the c-sections are supersolvable was discussed in [4]. [3] c-Sections, solvability and large subgroups of finite groups 293 We show in Proposition 2.8 that the (nonsolvable) group G = Aut(O'Nan) satisfies |Sec(M)| ≤ |G : M| β (with equality in some cases) for all maximal subgroups M of G. Thus β cannot be replaced by a larger constant in Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

“…The main result of [3] is that each group of composite order has a proper large subgroup. By applying Proposition 1.3 we prove the following theorem.…”

Section: Introductionmentioning

confidence: 99%

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c-SECTIONS, SOLVABILITY AND LARGE SUBGROUPS OF FINITE GROUPS

Baumeister¹,

Kaplan

2012

Bull. Aust. Math. Soc.

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c-Sections of maximal subgroups in a finite group and their relation to solvability have been extensively researched in recent years. A fundamental result due to Wang ['C-normality of groups and its properties', J. Algebra 180 (1998), [954][955][956][957][958][959][960][961][962][963][964][965] is that a finite group is solvable if and only if the c-sections of all its maximal subgroups are trivial. In this paper we prove that if for each maximal subgroup of a finite group G, the corresponding c-section order is smaller than the index of the maximal subgroup, then each composition factor of G is either cyclic or isomorphic to the O'Nan sporadic group (the converse does not hold). Furthermore, by a certain 'refining' of the latter theorem we obtain an equivalent condition for solvability. Finally, we provide an existence result for large subgroups in the sense of Lev ['On large subgroups of finite groups' J. Algebra 152 (1992), 434-438].2010 Mathematics subject classification: primary 20E34; secondary 20E28, 20D05, 20D10.

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