OD-characterization of all finite nonabelian simple groups with orders having prime divisors at most 13

The main purpose of this article is to determine h _OD (M) for every finite nonabelian simple group M with order having prime divisors at most 13. This result is an analog of the result by A. V. Vasil’ev [1] about the recognizability of these simple groups by spectrum (the set of element orders). By the available results, we need only consider the groups L ₆(3), U ₄(5), G ₂(4), L ₅(3), S ₄(8), U ₆(2), and O ₈ ⁺ (3).     

On recognition of finite simple groups with connected prime graph

The spectrum of a finite group is the set of its element orders. We prove a theorem on the structure of a finite group whose spectrum is equal to the spectrum of a finite nonabelian simple group. The theorem can be applied to solving the problem of recognizability of finite simple groups by spectrum.     

On the composition factors of a group with the same prime graph as B n (5)

Let G be a finite group. The prime graph of G is a graph whose vertex set is the set of prime divisors of |G| and two distinct primes p and q are joined by an edge, whenever G contains an element of order pq. The prime graph of G is denoted by Γ(G). It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if G is a finite group such that Γ(G) = Γ(B _n (5)), where n ? 6, then G has a unique nonabelian composition factor isomorphic to B _n (5) or C _n (5).     

Groups with the same prime graph as the orthogonal group B n (3)

Let G be a finite group. The prime graph of G is denoted by Γ(G). It is proved in [1] that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G ? B _p (3) or C _p (3). In this paper we prove the main result that if G is a finite group such that Γ(G) = Γ(B _n (3)), where n ≥ 6, then G has a unique nonabelian composition factor isomorphic to B _n (3) or C _n (3). Also if Γ(G) = Γ(B ₄(3)), then G has a unique nonabelian composition factor isomorphic to B ₄(3), C ₄(3), or ² D ₄(3). It is proved in [2] that if p is an odd prime, then B _p (3) is recognizable by element orders. We give a corollary of our result, generalize the result of [2], and prove that B _2k+1(3) is recognizable by the set of element orders. Also the quasirecognition of B _2k (3) by the set of element orders is obtained.     

On finite groups with disconnected prime graph

All finite simple nonabelian groups that have the same prime graph as a Frobenius group or a 2-Frobenius group are found.     

Finite groups having the same prime graph as the group <Emphasis Type=Italic>A</Emphasis><Subscript>10</Subscript>

We study solutions to partial differential equations with homogeneous polynomial symbols with respect to a multiplicative one-parameter transformation group such that all eigenvalues of the infinitesimal matrix are positive. The infinitesimal matrix may contain a nilpotent part. In the asymptotic scale of regularly varying functions, we find conditions under which such differential equations have asymptotically homogeneous solutions in the critical case.     

On nilpotent groups having an almost regular automorphism of prime order

Translated from Sibirskii Matematicheskii, Vol. 35, No. 3, pp. 630–632, May–June, 1994.     

On nonabelian composition factors of a finite prime spectrum minimal group

Assume that G is a primitive permutation group on a finite set X, x ∈ X, y ∈ X \ {x}, and G _x,y \(\underline \triangleleft \) G _x . P. Cameron raised the question about the validity of the equality G _x,y = 1 in this case. The author proved earlier that, if soc(G) is not a direct power of an exceptional group of Lie type, then G _x,y = 1. In the present paper, we prove that, if soc(G) is a direct power of an exceptional group of Lie type distinct from E ₆(q), ² E ₆(q), E ₇(q), and E ₈(q), then G _x,y = 1.     

On the structure of finite groups isospectral to an alternating group

It is proved that every finite group isospectral to an alternating group A _n of degree n greater than 21 has a chief factor isomorphic to an alternating group A _k , where k ≤ n and the half-interval (k, n] contains no primes.     

Cocliques of maximal size in the prime graph of a finite simple group

A prime graph of a finite group is defined in the following way: the set of vertices of the graph is the set of prime divisors of the order of the group, and two distinct vertices r and s are joined by an edge if there is an element of order rs in the group. We describe all cocliques of maximal size for finite simple groups.     

Finite groups having the same prime graph as the group Aut(<Emphasis Type="Italic">J</Emphasis><Subscript>2</Subscript>)

Finite groups having the same prime graph as the group Aut(J ₂) are described. This solves the problem posed by B. Khosravi.     

Covering theorems for nonabelian simple groups. II

In this part II, I study the class C of an n-cycle or of an n ? 1-cycle in the alternating group A_n. For n = 4k ? 1, 4k, CCC covers A_n, but CC does not. For n = 4k + 1, CCC covers A_n; I do not know whether CC does. For n = 4k + 2, CC covers A_n [Part III concerns PSL(2, q).]     

On alternating and symmetric groups as Galois groups

Fix an integern≧3. We show that the alternating groupA _n appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same whenn is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in anS _n-extension (i.e. a Galois extension with the symmetric groupS _n as Galois group). Forn≠6, it will follow thatA _n has the so-called GAR-property over any field of characteristic different from 2. Finally, we show that any polynomialf=X ⁿ+…+a₁X+a₀ with coefficients in a Hilbertian fieldK whose characteristic doesn’t dividen(n-1) can be changed into anS _n-polynomialf ^* (i.e the Galois group off ^* overK Gal(f ^*, K), isS _n) by a suitable replacement of the last two coefficienta ₀ anda ₁. These results are all shown using the Newton polygon. The author acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00114, GTEM.     

Recognition of finite groups by the prime graph

We obtain the first example of an infinite series of finite simple groups that are uniquely determined by their prime graph in the class of all finite groups. We also show that there exist almost simple groups for which the number of finite groups with the same prime graph is equal to 2. Supported by RFBR grant No. 05-01-00797, and by SB RAS Young Researchers Support grant No. 29 and Integration project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 390–408, July–August, 2006.     

Finite groups with a five-component prime graph

Some classification is obtained for the finite groups whose prime graph has five connected components. In particular, we show that such a group is simple. Some related results are stated about representations of simple groups.     

On the automorphism groups of symmetric graphs admitting an almost simple group

This paper investigates the automorphism group of a connected and undirected G-symmetric graph Γ where G is an almost simple group with socle T. First we prove that, for an arbitrary subgroup M of containing G, either T is normal in M or T is a subgroup of the alternating group A_k of degree . Then we describe the structure of the full automorphism group of G-locally primitive graphs of valency d, where d≤20 or is a prime. Finally, as one of the applications of our results, we determine the structure of the automorphism group for cubic symmetric graph Γ admitting a finite almost simple group.