Given a finite group G, let π (G) denote the set of all primes that divide the order of G. For a prime r ∈π (G) , we define r-singular elements as those elements of G whose order is divisible by r. Denote by S_r(G) the number of r-singular elements of G. We denote the proportion S_r(G)/|G| of r-singular elements in G by μ _r(G) . Let μ (G):= {μ _r(G) | r∈π (G)} be the set of all proportions of r-singular elements for each prime r in π (G) . In this paper, we prove that if a finite group G has the same set μ (G) as the simple group PSL(2,q) , then G is isomorphic to PSL(2,q) .
In this study, we investigate the existence and uniqueness of periodic solutions for a class of non-autonomous differential equations. By employing the perturbation property of semi-Fredholm operators with some suitable fixed point theorems, we confirm the periodicity of solutions on the positive real half-line under specific conditions on the linear and nonlinear parts of the equation. Moreover, we apply our findings to the diffusive logistic model with finite delay, accompanied by numerical simulations to clarify the practical relevance of our theoretical results.
We prove that the accumulation points of the approximate point spectrum of a bounded linear operator are invariant under commuting Riesz perturbations with finite spectrum.
