Spectra of finite symplectic and orthogonal groups

We give an affirmative answer to Question 12.39 in the Kourovka Notebook. Namely, it is proved that a finite simple group and a finite group having equal orders and same sets of element orders are isomorphic.The spectrum of a group G is the set ω(G) of its element orders. The spectrum of a finite group G together with its order retains a substantial part of information on the structure of G but, as demonstrated by the example of the dihedral group D 8 of order 8 and the quaternion group Q 8 , does not necessarily determine G uniquely. In [1], W. Shi conjectured that the desired uniqueness would be achieved for finite simple groups. In [2, Question 12.39], A. S. Kondratiev worded this conjecture as follows:Is it true that a finite group and a finite simple group are isomorphic if they have equal orders and same sets of element orders?Later, for brevity, it was suggested to refer to a finite group G that is isomorphic to every finite group H with ω(H) = ω(G) and |H| = |G| as recognizable by spectrum and order. In this terminology, Shi's question reads as follows:Is it true that all finite simple groups are recognizable by spectrum and order?The answer to this question is obviously 'yes' for Abelian simple groups. In a series of papers [1,[3][4][5][6][7][8], an affirmative answer was given for all non-Abelian simple groups except the symplectic groups, orthogonal groups of odd dimension, and orthogonal groups of type D n with n even. In the present paper, we argue for the recognizability by spectrum and order for these remaining groups. Thus, Shi's conjecture is confirmed and the following theorem holds true.

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“…Notice that ω(D ε m (u)) ⊆ ω(B m (u)) because B m (u) contains a section isomorphic to D ε m (u), and ω(B m (u)) ⊆ ω(C m (u)) by [19,Cors. 1,3].…”

Section: Preliminariesmentioning

confidence: 96%

Characterization of the finite simple groups by spectrum and order

2009

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“…Then C S (δ) 2 D 4 ( √ q) (see Proposition 4.9.2 of [19]). The group 2 D 4 ( √ q) contains an element of order q( √ q − 1) (see [12]); thus, 2(q √ q − 1) ∈ ω(G). This is impossible, as we showed above.…”

Section: Lemma 35 Suppose That S Is a Group Of Lie Type Over A Fielmentioning

confidence: 99%

“…A full description of the spectra for all finite simple symplectic and orthogonal groups is obtained in [12]. For B 3 (q), C 3 (q), and D 4 (q), the description depends on the parity of q and yields…”

Section: Properties Of Simple Symplectic and Orthogonal Groupsmentioning

confidence: 99%

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On recognition by spectrum of the simple groups B 3(q), C 3(q), and D 4(q)

Staroletov

2012

Sib Math J

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The spectrum of a finite group is the set of its element orders. Two groups are isospectral whenever they have the same spectra. We consider the classes of finite groups isospectral to the simple symplectic and orthogonal groups B 3 (q), C 3 (q), and D 4 (q). We prove that in the case of even characteristic and q > 2 these groups can be reconstructed from their spectra up to isomorphisms. In the case of odd characteristic we obtain a restriction on the composition structure of groups of this class.Keywords: finite group, simple symplectic and orthogonal groups, spectrum of a group, recognition by spectrum Consider a finite group G. The spectrum of ω(G) is the set of the element orders of G. It is not difficult to observe that the spectrum of a group is completely determined by the set μ(G) consisting of all divisibility-maximal elements of ω(G). Two groups are isospectral if their spectra coincide. We say that G is recognizable (by spectrum) if every group with the same spectrum as G is isomorphic to G.Given a group G, denote by h(G) the number of the isomorphism classes of the groups isospectral to G. Therefore, the recognizable groups are precisely G with h(G) = 1. Call a group G almost recognizable (by spectrum)Since every finite group with a nontrivial solvable normal subgroup is unrecognizable (see Lemma 1 of [1] for instance), the question of finding h(G) is interesting mainly when G is a simple or almost simple group (a group G is called almost simple whenever L ≤ G ≤ Aut(L) for some nonabelian simple group L). The most complete survey is [1] of groups for which this question is answered. In particular, the survey indicates that ω(D 4 (2)) = ω(B 3 (2)) and h(D 4 (2)) = h(B 3 (2)) = 2 (see [2]). In addition, C 3 (3) is recognizable (see [3]), while D 4 (3) and B 3 (3) are isospectral; furthermore, h(D 4 (3)) = 2 (see [2]). The main result of [4] implies that C 3 (4) and D 4 (4) are recognizable groups.This article is devoted to studying recognition for B 3 (q), C 3 (q), and D 4 (q). Namely, we prove Theorem 1. Suppose that L is one of the simple groups B 3 (q), C 3 (q), or D 4 (q), where q is a power of a prime p and q ≥ 5. If S is a composition factor of a finite group isospectral to L and S is isomorphic to a group of Lie type over a field of characteristic p then S ∈ {B 3 (q), C 3 (q), D 4 (q)}.Recall that B 3 (q) and C 3 (q) are isomorphic for even q.Theorem 2. C 3 (q) and D 4 (q) are recognizable by spectrum for even q > 2. If q = 2 then C 3 (2) and D 4 (2) are isospectral, and h(C 3 (2)) = 2.

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“…Clearly, the time required for this verification is bounded by a polynomial in k log M , where k = |ν(G)|. Using the description of spectra of simple groups (see, e.g., [6,7] for the case of classical groups), we may construct some set ν(G) for every simple group G. The question whether the cardinality of this set can be bounded polynomially in terms of m remains open.…”

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confidence: 99%

On Constructive Recognition of Finite Simple Groups by Element Orders

Buturlakin

Васильев

2014

Algebra Logic

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The set of element orders of a finite group G is called the spectrum and is denoted by ω(G), and groups with the same spectrum are said to be isospectral. The following question seems to be natural: If M is a set of positive integers, does then a finite group G with ω(G) = M exist, and if so, can we describe all such groups? The paper is concerned with an algorithmic aspect of this problem in the special case where G is simple. The last restriction is justified by the following reasons. Obviously, there exist nonisomorphic isospectral finite groups; moreover, if G is a finite group with nontrivial soluble radical, then we can construct infinitely many pairwise nonisomorphic finite groups isospectral to G (see, e.g., [1]). On the other hand, in general, the set of groups isospectral to a finite non-Abelian simple group G is finite and consists of groups closely related to G (see, e.g., [2,3]). Thus if the name (according to CFSG) of a simple group G with ω(G) = M is determined, then normally a complete list of groups enjoying this property can be created.Note that the spectrum of a group G is closed under taking divisors; i.e., if n ∈ ω(G) and d divides n then d ∈ ω(G). In particular, ω(G) is uniquely determined by its subset μ(G) of elements maximal w.r.t. divisibility. Let ω(M) and μ(M) stand for respectively the set of all divisors of elements of M and the set of all elements of M maximal w.r.t. divisibility. If we require ω(G) = ω(M) instead of ω(G) = M, then we obtain a more natural problem whose solution obviously implies a solution to the original problem.

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Spectra of finite symplectic and orthogonal groups

Cited by 41 publications

References 6 publications

Characterization of the finite simple groups by spectrum and order

Characterization of the finite simple groups by spectrum and order

On recognition by spectrum of the simple groups B 3(q), C 3(q), and D 4(q)

On Constructive Recognition of Finite Simple Groups by Element Orders

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