‎There are a few finite groups that are determined up to isomorphism solely by their order, such as $mathbb{Z}_{2}$ or $mathbb{Z}_{15}$. Still other finite groups are determined by their order together with other data, such as the number of elements of each order, the structure of the prime graph, the number of order components, the number of Sylow $p$-subgroups for each prime $p$, etc. In this...

‎let $g$ be a finite group‎. ‎in [ghasemabadi et al.‎, ‎characterizations of the simple group ${}^2d_n(3)$ by prime graph‎ ‎and spectrum‎, ‎monatsh math.‎, ‎2011] it is‎ ‎proved that if $n$ is odd‎, ‎then ${}^2d _n(3)$ is recognizable by‎ ‎prime graph and also by element orders‎. ‎in this paper we prove‎ ‎that if $n$ is even‎, ‎then $d={}^2d_{n}(3)$ is quasirecognizable by‎ ‎prime graph‎, ‎i.e‎...

1 Multiply transitive groups Theorem 1.1. Let Ω be a finite set and G ≤ Sym(Ω) be 2–transitive. Let N E G be a minimal normal subgroup. Then one of the following holds: (a) N is regular and elementary abelian. (b) N is primitive, simple and not abelian. Proof. First we show that N is unique. Suppose that M is another minimal normal subgroup of G, so N ∩M = {e} and therefore [N,M ] = {e}. Since ...