let $pi_e(g)$ be the set of element orders of a finite group $g$‎. ‎let $nse(g)={m_nmid ninpi_e(g)}$‎, ‎where $m_n$ be the number of elements of order $n$ in $g$‎. ‎in this paper‎, ‎we prove that if $nse(g)=nse(l_2(81))$‎, ‎then $gcong l_2(81)$‎.

in this paper we show that if q is a power of a prime p , then the projective special linear group psl(2, q) and the stabilizer of a point of the projective line have maximum sum element orders among all proper subgroups of projective general linear group pgl(2, q) for q odd and even respectively

Saving a patients live in an emergency state especially if he is far from medical institutions, is considered to be the primary concern of the physician. With a mobile physician, patients health state may vary to be in a critical situation that requires immediate medical intervention. Having chosen the medical institution as destination, the doctor must react immediately and reach his patient a...

i.e., the supremum of all orders of torsion points on elliptic curves defined over some degree d number field. Write T (d)′ for the set of prime divisors of elements of Td, and P (d) for the largest element of T (d)′. Let TCM(d) (resp. TIM(d)) be the subset of T (d) corresponding to elliptic curves with complex multiplication (resp. with algebraic integral modulus j(E)), and similarly adding th...

Given a rectangle A and a set S of n points in A, we consider the problem, called the maximum empty rectangle problem, of finding a maximum area rectangle that is fully contained in A and does not contain any point of S in its interior. An O(n’) time algorithm is presented. Furthermore, it is shown that if the points of S are drawn randomly and independently from A, the problem can be solved in...

Let $G$ be a finite group and $\psi(G) = \sum_{g \in G} o(g)$, where $o(g)$ denotes the order of $g G$. First, we prove that if is $n$ >31\psi(C_n)/77$, $C_n$ cyclic $n$, then supersolvable. This proves conjecture M.~{T\u{a}rn\u{a}uceanu}. Moreover, M. Herzog, P. Longobardi Maj put forward following conjecture: If $H\leq G$, \leqslant \psi(H) |G:H|^2$. In sequel, by an example show this not sat...

Let $ G be a periodic group, and let \omega(G)\subseteq{