Recognition by prime graph of the almost simple group PGL(2, 25) (original) (raw)
Related papers
arXiv: Group Theory, 2016
The prime graph of a finite group GGG is denoted by ga(G)\ga(G)ga(G). Also GGG is called recognizable by prime graph if and only if each finite group HHH with ga(H)=ga(G)\ga(H)=\ga(G)ga(H)=ga(G), is isomorphic to GGG. In this paper, we classify all finite groups with the same prime graph as textrmPGL(2,9)\textrm{PGL}(2,9)textrmPGL(2,9). In particular, we present some solvable groups with the same prime graph as textrmPGL(2,9)\textrm{PGL}(2,9)textrmPGL(2,9).
Quasirecognition by prime graph of the simple group Ln (3)
q) (q = 3 2n+1 ) or 2 B 2 (q) (q = 2 2n+1 > 2), then G is quasirecognizable by prime graph. Hence we generalize some known results of G 2 (q) and 2 B 2 (q). A finite simple nonabelian group G is said to be quasirecognizable by spectrum if each finite group H with ω(H) = ω(G) has a composition factor isomorphic to G (see ).
Quasirecognition by prime graph of finite simple groups L n (2) and U n (2)
Acta Mathematica Hungarica, 2011
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.
A characterization of the finite simple group L 16 (2) by its prime graph
Manuscripta Mathematica, 2008
In this paper as the main result we prove that the projective special linear group L 16(2) is uniquely determined by its prime graph. In fact we give a positive answer to an open problem arose in Zavarnitsin (Algebra Logic 43(4), 220–231, 2006) and we obtain a first example of a finite group with connected prime graph which is uniquely determined by its prime graph.
On some Frobenius groups with the same prime graph as the almost simple group PGL(2,49)
arXiv: Group Theory, 2016
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.
Quasirecognition by prime graph of the simple group 2 G 2 ( q )
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.
Recognition of Finite Simple Groups Whose First Prime Graph Components Are г-Regular
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].
Quasirecognition by Prime Graph of 2 Dp(3) Where p = 2n + 1 5 is a Prime
In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ( 2 D p (3)), where p = 2 n + 1, (n ≥ 2) is a prime number, then G has a unique non-abelian composition factor isomorphic to 2 D p (3). We also show that if G is a finite group satisfying |G| = | 2 D p (3)| and Γ(G) = Γ( 2 D p (3)), then G ∼ = 2 D p (3). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for 2 D p (3). Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered.