Recognizability of Symmetric Groups by Spectrum

Gorshkov, Ilya

doi:10.1007/s10469-015-9306-0

Cited by 6 publications

(7 citation statements)

References 14 publications

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“…It is proved in [1] that if L = A n , where n ≥ 5 with n = 6, 10, and G is a finite group such with ω(G) = ω(L) then G ≃ L. If L = S n , then the same conclusion is established for n = 2, 3, 4, 5, 6, 8, 10, see [2] and [3]. Therefore, for almost all n the groups A n and S n are characterized by spectrum in the class of finite groups.…”

Section: Introductionmentioning

confidence: 81%

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On groups having the prime graph as alternating and symmetric groups

Gorshkov

Staroletov

2019

Communications in Algebra

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The prime graph Γ(G) of a finite group G is the graph whose vertex set is the set of prime divisors of |G| and in which two distinct vertices r and s are adjacent if and only if there exists an element of G of order rs. Let An (Sn) denote the alternating (symmetric) group of degree n. We prove that if G is a finite group with Γ(G) = Γ(An) or Γ(G) = Γ(Sn), where n ≥ 19, then there exists a normal subgroup K of G and an integer t such that At ≤ G/K ≤ St and |K| is divisible by at most one prime greater than n/2. * The work was supported by the program of fundamental scientific researches of the SB RAS

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Section: Introductionmentioning

confidence: 81%

“…Lemma 4. (Zsigmondy [8]) Given an integer a with |a| > 1, for every positive integer i the set R i (a) is nonempty except for the pairs (a, i) ∈ {(2, 1), (2,6), (−2, 2), (−2, 3), (3, 1), (−3, 2)}.…”

Section: Introductionmentioning

confidence: 99%

On groups having the prime graph as alternating and symmetric groups

Gorshkov

Staroletov

2019

Communications in Algebra

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“…It is proved in [11] Grunberg-Kegel graph (or prime graph) GK(G) of group G is defined as follows. The set of vertices of this graph is π(G).…”

Section: Examples and Theoremsmentioning

confidence: 99%

“…Until now, all the considered symmetric groups, with the exception of group S9, the recognizability of which was proved, had a disconnected prime graph. A symmetric group of degree n>2 has a disconnected prime graph only if one of the numbers n or n-1 is prime [11].…”

Section: Examples and Theoremsmentioning

confidence: 99%

On Recognizability of Groups by Bottom Layer

Senashov¹,

Parashchuk²

2020

AMA_A

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The bottom layer of a group is a set of its elements of prime order. A group is called recognizable by bottom layer under additional conditions if it is uniquely restored by bottom layer under these conditions. A group is called almost recognizable by bottom layer under additional conditions, if there are a finite number of pairwise non-isomorphic groups satisfying these conditions, with bottom layer that is the same as that of the group. A group is called unrecognizable by bottom layer under additional conditions if there are an infinite number of pairwise non-isomorphic groups satisfying these conditions, with bottom layer that is the same as that of the group. In the paper we consider examples of groups recognized by bottom layer, by spectrum and, simultaneously, by spectrum and by bottom layer. We have also proved some results of recognizability of groups by bottom layer.

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“…for some nonabelian simple groups L i , 1 i k. Assume first that k = 1 in (4.1), that is, G is an almost simple group. The recognition problem is solved for all symmetric groups Sym n except Sym 10 (see [24,26] and references therein). Namely, they are unrecognizable if n ∈ {2, 3, 4, 5, 6, 8} and recognizable otherwise.…”

Section: The Group ∪ ∞mentioning

confidence: 99%

On the structure of finite groups isospectral to finite simple groups

Grechkoseeva

Vasil’ev

2015

Journal of Group Theory

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Finite groups are said to be isospectral if they have the same sets of element orders. A finite nonabelian simple group $L$ is said to be almost recognizable by spectrum if every finite group isospectral to $L$ is an almost simple group with socle isomorphic to $L$. It is known that all finite simple sporadic, alternating and exceptional groups of Lie type, except $J_2$, $A_6$, $A_{10}$ and $^3D_4(2)$, are almost recognizable by spectrum. The present paper is the final step in the proof of the following conjecture due to V.D. Mazurov: there exists a positive integer $d_0$ such that every finite simple classical group of dimension larger than $d_0$ is almost recognizable by spectrum. Namely, we prove that a nonabelian composition factor of a~finite group isospectral to a finite simple symplectic or orthogonal group $L$ of dimension at least 10, is either isomorphic to $L$ or not a group of Lie type in the same characteristic as $L$, and combining this result with earlier work, we deduce that Mazurov's conjecture holds with $d_0=60$.Comment: 13 page

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Recognizability of Symmetric Groups by Spectrum

Cited by 6 publications

References 14 publications

On groups having the prime graph as alternating and symmetric groups

On groups having the prime graph as alternating and symmetric groups

On Recognizability of Groups by Bottom Layer

On the structure of finite groups isospectral to finite simple groups

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