Degrees of faithful irreducible representations of metabelian groups

In 1993, Sim proved that all the faithful irreducible representations of a finite metacyclic group over any field of positive characteristic have the same degree. In this paper, we restrict our attention to non-modular representations and generalize this result for — (1) finite metabelian groups, over fields of positive characteristic coprime to the order of groups, and (2) finite groups having a cyclic quotient by an abelian normal subgroup, over number fields.

Some Approach to Key Exchange Protocol Based on Non-commutative Groups

We consider non-commutative generalization of CDH problem [1,2] on base of metacyclic group G of type Millera Moreno (minimal non-abelian group). We show that conjugacy problem in this group are intractable. The algorithm of generating (desinging) of common key in non-commutative group with 2 mutually commuting subgroups are constructed by us.

Bound on the diameter of metacyclic groups

Let [Formula: see text] be the split metacyclic group, where [Formula: see text] is a unit modulo [Formula: see text]. We derive an upper bound for the diameter of [Formula: see text] using an arithmetic parameter called the weight, which depends on [Formula: see text], [Formula: see text], and the order of [Formula: see text]. As an application, we show how this would determine a bound on the diameter of an arbitrary metacyclic group.

Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4n

We construct several new families of directed strongly regular Cayley graphs (DSRCGs) over the metacyclic group M 4 n = ⟨ a , b | a n = b 4 = 1 , b − 1 a b = a − 1 ⟩ , some of which generalize those earlier constructions. For a prime p and a positive integer α > 1 , for some cases, we characterize the DSRCGs over M 4 p α .

A note on normal complement problem

In this paper, we study the normal complement problem on semisimple group algebras and modular group algebras [Formula: see text] over a field [Formula: see text] of positive characteristic. We provide an infinite class of abelian groups [Formula: see text] and Galois fields [Formula: see text] that have normal complement in the unit group [Formula: see text] for semisimple group algebras [Formula: see text]. For metacyclic group [Formula: see text] of order [Formula: see text], where [Formula: see text] are distinct primes, we prove that [Formula: see text] does not have normal complement in [Formula: see text] for finite semisimple group algebra [Formula: see text]. Finally, we study the normal complement problem for modular group algebras over field of characteristic 2.

Colorings, determinants and Alexander polynomials for spatial graphs

A balanced spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to S. Kinoshita, Alexander polynomials as isotopy invariants I, Osaka Math. J. 10 (1958) 263–271.), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and [Formula: see text]-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which [Formula: see text] the graph is [Formula: see text]-colorable, and that a [Formula: see text]-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group [Formula: see text]. We finish by proving some properties of the Alexander polynomial.