Affine difference algebraic groups are a generalization of affine obtained by replacing equations with equations. We show that the isomorphism theorems from abstract group theory have meaningful analogs for these and we establish Jordan–Hölder type theorem allows us to decompose any into almost-simple groups. also characterize via

let $g $ be a finite group and let $gamma(g)$ be the prime graph‎ ‎of g‎. ‎assume $2 < q = p^{alpha} < 100$‎. ‎we determine finite groups‎ ‎g such that $gamma(g) = gamma(u_3(q))$ and prove that if $q neq‎ ‎3‎, ‎5‎, ‎9‎, ‎17$‎, ‎then $u_3(q)$ is quasirecognizable by prime graph‎, ‎i.e‎. ‎if $g$ is a finite group with the same prime graph as the‎ ‎finite simple group $u_3(q)$‎, ‎then $g$ has a un...

we introduce the class of “right almost v-rings” which is properly between the classes of right v-rings and right good rings. a ring r is called a right almost v-ring if every simple r-module is almost injective. it is proved that r is a right almost v-ring if and only if for every r-module m, any complement of every simple submodule of m is a direct summand. moreover, r is a right almost v-rin...

Let G be a finite almost simple group with socle G0. A (nontrivial) factorization of is an expression the form G=HK, where factors H and K are core-free subgroups. There extensive literature on factorizations groups, important applications in permutation theory algebraic graph theory. In recent paper, Xia second-named author describe groups solvable factor H. Several infinite families arise con...