We prove that connected vertex-transitive digraphs of order p5 (where p is a prime) are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup G of prime power order such that G′ is generated by two elements or elementary abelian is Hamiltonian.

‎In the classical group theory there is‎ an open question‎: ‎Is every torsion free n-Engel group (for n ≥ 4)‎, nilpotent?‎. ‎To answer the question‎, ‎Traustason‎ [11] showed that with some additional conditions all‎ ‎4-Engel groups are locally nilpotent‎. ‎Here‎, ‎we gave some partial‎ answer to this question on Engel fuzzy subgroups‎. ‎We show that if μ is a normal 4-Engel fuzzy‎ subgroup of ...

In this paper, we present a characterization of groups [Formula: see text] order text], prime, in which not all elements the commutator subgroup are commutators text]. way, obtain several structural results on

For simplicity, we use the following convention: G is a group, a, b are elements of G, m, n are natural numbers, and p is a prime natural number. One can prove the following propositions: (1) If for every natural number r holds n 6= pr, then there exists an element s of N such that s is prime and s | n and s 6= p. (2) For all natural numbers n, m such that n | pm there exists a natural number r...

Methods for Avoiding the "Small-Subgroup" Attacks on the Diffie-Hellman Key Agreement Method for S/MIME Status of this Memo This memo provides information for the Internet community. It does not specify an Internet standard of any kind. Distribution of this memo is unlimited. Abstract In some circumstances the use of the Diffie-Hellman key agreement scheme in a prime order subgroup of a large p...

This paper gives a complete classification of the finite groups that contain a strongly closed p-subgroup for p any prime.

Uniformly describing the conjugacy classes of unipotent upper triangular groups \(\mathrm{UT}_{n}(\mathbb{F}_{q})\) (for all or many values \(n\) and \(q\)) is a nearly impossible task. This paper takes on related problem normal subgroups \(\mathrm{UT}_{n}(\mathbb{F}_{q})\). For \(q\) prime, bijection will be established between these pairs combinatorial objects with labels from \(\mathbb{F}_{q...

Let K be a global field of char p and let Fq be the algebraic closure of Fp in K. For an elliptic curve E/K with non-constant j-invariant, the L-function L(T,E/K) is a polynomial in 1+T ·Z[T ]. For any N > 1 invertible in K and finite subgroup T ⊂ E(K) of order N , we compute the mod N reduction of L(T,E/K) and determine an upper-bound for the order of vanishing at 1/q, the so-called analytic r...