Quasirecognition of E6(q) by the orders of maximal abelian subgroups

In [Li and Chen, A new characterization of the simple group [Formula: see text], Sib. Math. J. 53(2) (2012) 213–247.], it is proved that the simple group [Formula: see text] is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as [Formula: see text], Monatsh. Math. 174 (2013) 285–303], the authors proved that if [Formula: see text], where [Formula: see text] is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as [Formula: see text], is isomorphic to [Formula: see text] or an extension of [Formula: see text] by a subgroup of the outer automorphism group of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group with the same orders of maximal abelian subgroups as [Formula: see text], then [Formula: see text] has a unique nonabelian composition factor which is isomorphic to [Formula: see text].

On recognition by prime graph of the projective special linear group over GF(3)

Let G be a finite group. The prime graph of G is denoted by ?(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 ?(G) = ?(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.

QUASIRECOGNITION BY PRIME GRAPH OF SOME ORTHOGONAL GROUPS OVER THE BINARY FIELD

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 D = Dn(2), where n ≥ 3 or 2Dn(2), where n ≥ 15. In this paper we prove that D is quasirecognizable by prime graph, i.e. every finite group G with Γ(G) = Γ(D), has a unique nonabelian composition factor which is isomorphic to D. Finally, we consider the quasirecognition by spectrum for these groups. Specially we prove that if p = 2n + 1 ≥ 17 is a prime number, then Dp(2) is recognizable by spectrum.

ON THE BASE SIZE OF A TRANSITIVE GROUP WITH SOLVABLE POINT STABILIZER

We prove that the base size of a transitive group G with solvable point stabilizer and with trivial solvable radical is not greater than k provided the same statement holds for the group of G-induced automorphisms of each nonabelian composition factor of G.

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

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 there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

RECOGNITION OF SOME FINITE SIMPLE GROUPS OF TYPE Dn(q) BY SPECTRUM

The spectrum ω(G) of a finite group G is the set of element orders of G. Let L be finite simple group Dn(q) with disconnected Gruenberg–Kegel graph. First, we establish that L is quasi-recognizable by spectrum except D4(2) and D4(3), i.e., every finite group G with ω(G) = ω(L) has a unique nonabelian composition factor that is isomorphic to L. Second, for some special series of integers n, we prove that L is recognizable by spectrum, i.e., every finite group G with ω(G) = ω(L) is isomorphic to L.