Recognition by spectrum for finite linear groups over fields of characteristic 2

Journal of Group Theory

2017

We say that a finite almost simple G with socle S is admissible (with respect to the spectrum) if G and S have the same sets of orders of elements. Let L be a finite simple linear or unitary group of dimension at least three over a field of odd characteristic. We describe admissible almost simple groups with socle L. Also we calculate the orders of elements of the coset Lτ , where τ is the inverse-transpose automorphism of L.

“…Proof. If ε = +, the assertion is proved in [10,Propositions 6,7], and if ε = −, it is proved in [12,Proposition 6]. Proof.…”

Section: Admissible Groupsmentioning

confidence: 97%

On orders of elements of finite almost simple groups with linear or unitary socle

Journal of Group Theory

2017

“…The latest and most general result on recognition of linear groups over arbitrary finite fields of characteristic 2 was obtained by Grechkoseeva [8]. For natural numbers m and l, the l-part of m is the maximal divisor t of m such that π(t) ⊆ π(l).…”

Section: Theorem 11mentioning

confidence: 99%

Recognition by spectrum for finite simple groups of Lie type

Shi²,

Vasil’ev

2008

Front. Math. China

The goal of this article is to survey new results on the recognition problem. We focus our attention on the methods recently developed in this area. In each section, we formulate related open problems. In the last two sections, we review arithmetical characterization of spectra of finite simple groups and conclude with a list of groups for which the recognition problem was solved within the last three years.

“…Furthermore, they all are in ω(L). Let r i ∈ π(k i ), 3 i n. According to [19,Tables 4,8], the independence number t(L) is equal to [ Let G be a finite group and ω(G) = ω(L). By Lemma 1, G has a unique non-Abelian composition factor S. Denote the soluble radical of G by K. Then S ≤ G = G/K ≤ Aut S. Furthermore, S satisfies t(S) t(G) − 1, and any number in π(k n−1 ) ∪ π(k n ) does not divide the product |K| · |G/S|.…”

Section: Proof Of the Theoremmentioning

confidence: 99%

“…For simple linear groups L n (2 k ), the recognizability problem is solved with n = 2 [3], n = 3 [4,5], n = 4 [6], 11 n 17 [7,8], n 26 [8,9], and also for k = 1 [10,11]. The goal of the present paper is to solve the problem for all the remaining groups L n (2 k ), thus settling the question of whether finite simple linear groups over fields of characteristic 2 are recognizable by spectrum.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Recognition by spectrum for finite simple linear groups of small dimensions over fields of characteristic 2

Vasilyev

2008

Algebra Logic

Self Cite

Keywords: finite simple group, linear group, order of element, spectrum of group, recognition by spectrum.Two groups are said to be isospectral if they share the same set of element orders. For every finite simple linear group L of dimension n over an arbitrary field of characteristic 2, we prove that any finite group G isospectral to L is isomorphic to an automorphic extension of L. An explicit formula is derived for the number of isomorphism classes of finite groups that are isospectral to L. This account is a continuation of the second author's previous paper where a similar result was established for finite simple linear groups L in a sufficiently large dimension (n > 26), and so here we confine ourselves to groups of dimension at most 26.