ON RECOGNITION OF THE FINITE SIMPLE ORTHOGONAL GROUPS OF DIMENSION 2 m , 2 m + 1, AND 2 m + 2 OVER A FIELD OF CHARACTERISTIC 2

Abstract: The spectrum ω(G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable by spectrum (briefly, recognizable) if H G for every finite group H such that ω(H) = ω(G). We give two series, infinite by dimension, of finite simple classical groups recognizable by spectrum

On finite groups isospectral to . . .

Let L be a simple linear or unitary group of dimension larger than 3 over a finite field of characteristic p. We deal with the class of finite groups isospectral to L. It is known that a group of this class has a unique nonabelian composition factor. We prove that if L = U 4 (2), U 5 (2) then this factor is isomorphic either to L or a group of Lie type over a field of characteristic different from p

Criterion of nonsolvability of a finite group and recognition of direct squares of simple groups

The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if among the prime divisors of the order of a group $G$, there are four different primes such that $\omega(G)$ contains all their pairwise products but not a product of any three of these numbers, then $G$ is nonsolvable. Using this result, we show that for $q\geqslant 8$ and $q\neq 32$, the direct square $Sz(q)\times Sz(q)$ of the simple exceptional Suzuki group $Sz(q)$ is uniquely characterized by its spectrum in the class of finite groups, while for $Sz(32)\times Sz(32)$, there are exactly four finite groups with the same spectrum.Comment: In the third version, Theorem 1 is slightly reformulated and some references are correcte