On The Unrecognizability by Prime Graph for the Most Simple Group PGL(2,9){\rm {\bf PGL(2,9)}}PGL(2,9) (original) (raw)
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Throughout this paper, every groups are nite. The prime graph of a group G is denoted by ( G). Also G is called recognizable by prime graph if for every nite group H with ( H) = ( G), we conclude that GH. Until now, it is proved that if k is an odd number and p is an odd prime number, then PGL(2;p k ) is recognizable by prime graph. So if k is even, the recognition by prime graph of PGL(2;p k ), where p is an odd prime number, is an open problem. In this paper, we generalize this result and we prove that the almost simple group PGL(2;25) is recognizable by prime graph. c
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Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if G has an element of order pp′. Let L=L n (2) or U n (2), where n≧17. We prove that L is quasirecognizable by prime graph, i.e. if G is a finite group such that Γ(G)=Γ(L), then G has a unique nonabelian composition factor isomorphic to L. As a consequence of our result we give a new proof for the recognition by element orders of L n (2). Also we conclude that the simple group U n (2) is quasirecognizable by element orders.
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The prime graph of a finite group GGG is denoted by ga(G)\ga(G)ga(G) whose vertex set is pi(G)\pi(G)pi(G) and two distinct primes ppp and qqq are adjacent in ga(G)\ga(G)ga(G), whenever GGG contains an element with order pqpqpq. We say that GGG is unrecognizable by prime graph if there is a finite group HHH with ga(H)=ga(G)\ga(H)=\ga(G)ga(H)=ga(G), in while HnotcongGH\not\cong GHnotcongG. In this paper, we consider finite groups with the same prime graph as the almost simple group textrmPGL(2,49)\textrm{PGL}(2,49)textrmPGL(2,49). Moreover, we construct some Frobenius groups whose their prime graph coincide with ga(textrmPGL(2,49))\ga(\textrm{PGL}(2,49))ga(textrmPGL(2,49)), in particular, we get that textrmPGL(2,49)\textrm{PGL}(2,49)textrmPGL(2,49) is unrecognizable by prime graph.
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Let G be a finite group and cd(G) denote the character degree set for G. The prime graph ∆(G) is a simple graph whose vertex set consists of prime divisors of elements in cd(G), denoted ρ(G). Two primes p, q ∈ ρ(G) are adjacent in ∆(G) if and only if pq|a for some a ∈ cd(G). We determine which simple 4-regular graphs occur as prime graphs for some finite nonsolvable group. AMS subject classification: 20C15
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Acta Mathematica Hungarica, 2009
Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ(2G2(q)), where q = 32n+1 for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to 2G 2(q). We infer that if G is a finite group satisfying |G| = |2G 2(q)| and Γ(G) = Γ (2G 2(q)) then G ≅ = 2G 2(q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.
Journal of Linear and Topological Algebra, 2017
The prime graph of a finite group G is denoted by Γ(G) whose vertex set is π(G) and two distinct primes p and q are adjacent in Γ(G), whenever G contains an element with order pq. We say that G is unrecognizable by prime graph if there is a finite group H with Γ(H) = Γ(G), in while H ̸ ∼ = G. In this paper, we consider finite groups with the same prime graph as the almost simple group PGL(2, 49). Moreover, we construct some Frobenius groups whose prime graphs coincide with Γ(PGL(2, 49)), in particular, we get that PGL(2, 49) is unrecognizable by its prime graph.
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Publications de l'Institut Mathematique, 2014
Let G be a finite group. The prime graph of G is denoted by Γ(G). We prove that the simple group PSLn(3), where n 9, is quasirecognizable by prime graph; i.e., if G is a finite group such that Γ(G) = Γ(PSLn(3)), then G has a unique nonabelian composition factor isomorphic to PSLn(3). Darafsheh proved in 2010 that if p > 3 is a prime number, then the projective special linear group PSLp(3) is at most 2-recognizable by spectrum. As a consequence of our result we prove that if n 9, then PSLn(3) is at most 2-recognizable by spectrum.
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Let G be a finite group and π(G) = {p 1 , p 2 , • • • , p s }. For p ∈ π(G), we put deg(p) : = |{q ∈ π(G)|p ∼ q in the prime graph of G}|, which is called the degree of p. We also define D(G) := (deg(p 1), deg(p 2),. .. , deg(p s)), where p 1 < p 2 < • • • < p s , which is called the degree pattern of G. We say G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as G. In particular, a 1-fold OD-characterizable group is simply called an OD-characterizable group (see [13]). In the present paper, we determine all finite simple groups whose first prime graph components are 1-regular and prove that all finite simple groups whose first prime graph components are r-regular except U 4 (2) are OD-characterizable, where 0 ≤ r ≤ 2. In particular, U 4 (2) is exactly 2-fold ODcharacterizable, which improves a result in [25].