The Algebraicity Conjecture treats model-theoretic foundations of algebraic group theory. It states that any simple group of finite Morley rank is an algebraic group over an algebraically closed field. In the mid-1990s a view was consolidated that this project falls into four cases of different flavour: even type, mixed type, odd type, and degenerate type. This book contains a proof of the conj...

‎Suppose G is a group of order p^2q^2 where p>q are prime numbers and suppose P and Q are Sylow p-subgroups and Sylow q-subgroups of G, ‎respectively‎. ‎In this paper‎, ‎we show that up to isomorphism‎, ‎there are four groups of order p^2q^2 when Q and P are cyclic‎, ‎three groups when Q is a cyclic and P is an elementary ablian group‎, ‎p^2+3p/2+7 groups when Q is an elementary ablian group an...

‎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...