Definition 1. We say that the partial order P is a projection of the partial order Q and denote this by P Q, if there is an onto mapping π : Q→ P which is order preserving and such that ∀q ∈ Q∀p ∈ P s.t. π(q) ≤ p there is q′ ∈ Q (q ≤Q q′) ∧ (π(q) = p). Furthermore whenever π(q) ≤ p there is a condition q1 in Q which is usually denoted p + q such that q ≤ q1 and for every r ∈ Q such that p ≤ π(r...

in this article, we prove the existence of extremal positive solution for the distributed order fractional hybrid differential equation$$int_{0}^{1}b(q)d^{q}[frac{x(t)}{f(t,x(t))}]dq=g(t,x(t)),$$using a fixed point theorem in the banach algebras. this proof is given in two cases of the continuous and discontinuous function $g$, under the generalized lipschitz and caratheodory conditions.

Let $p=(q^4+q^3+q^2+q+1)/(5,q-1)$ be a prime number, where $q$ is a prime power. In this paper, we will show $Gcong mathrm{PSL}(5,q)$ if and only if $|G|=|mathrm{PSL}(5,q)|$, and $G$ has a conjugacy class size $frac{| mathrm{PSL}(5,q)|}{p}$. Further, the validity of a conjecture of J. G. Thompson is generalized to the groups under consideration by a new way.