Let G be a finite group. An irreducible character χ is called monolithic when the factor group G/ker⁡(χ) has unique minimal normal subgroup. In this paper, we prove that for smallest prime q dividing order of if faithful imprimitive degree q, then becomes nonabelian q-group or Frobenius with cyclic complement whose q. Under certain conditions, also classify groups in which their nonlinear chara...

New family of flat potential (Darboux-Egoroff) metrics on the Hurwitz spaces and corresponding Frobenius structures are found. We consider a Hurwitz space as a real manifold. Therefore the number of coordinates is twice as big as the number of coordinates used in the construction of Frobenius manifolds on Hurwitz spaces found by B.Dubrovin more than 10 years ago. The branch points of a ramified...

Abstract We prove that if the Auslander–Reiten triangles generate relations for Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with (co)generator, then has only finitely many isomorphism classes indecomposable objects up to translation. This gives converse theorem Butler and on groups. Our approach applications in context Frobenius categories.

A 2-Sylow subgroup of J is elementary abelian of order 8 and J has no subgroup of index 2. If r is an involution in J, then C(r) = (r) X K, where K _ A5. Let G be a finite group with the following properties: (a) S2-subgroups of G are abelian; (b) G has no subgroup of index 2; and (c) G contains an involution t such that 0(t) = (t) X F, where F A5. Then G is a (new) simple group isomorphic to J...

In [1], the multiple Frobenius-Euler numbers and polynomials were constructed. In this paper we give some interesting formulae which are related to the multiple Frobenius-Euler polynomials. The main purpose of this paper is to give the Kummer type congruences for the multiple Frobenius-Euler numbers. §

Iskander Aliev A sharp lower bound for the Frobenius number Ferdinand Georg Frobenius (1849–1917) raised the following problem: given N positive integers a1, . . . , aN with gcd(a1, . . . , aN ) = 1, find the largest natural number gN = gN (a1, . . . , aN) (called the Frobenius

This is the introductory chapter to this issue. We review the main idea of the localization technique and its brief history both in geometry and in QFT. We discuss localization in diverse dimensions and give an overview of the major applications of the localization calculations for supersymmetric theories. We explain the focus of the present issue.

We construct a pair of non-commutative rank 8 association schemes from 3 non-symmetric scheme. For the pair, two have same character table but different Frobenius-Schur indicators. This situation is similar to dihedral group and quaternion order 8. also determine structures adjacency algebras them over rational number field.

We establish a new universal relation between the Lie bracket and • –multiplication of tangent fields on any Frobenius (super)manifold. We use this identity in order to introduce the notion of " weak Frobenius manifold " which does not involve metric as part of structure. As another application, we show that the powers of an Euler field generate (a half of) the Virasoro algebra on an arbitrary,...

Let C be a smooth curve, and Mr(C) the coarse moduli space of vector bundles of rank r and trivial determinant on C. We discuss the generalized Verschiebung map Vr : Mr(C) 99K Mr(C) induced by pulling back under Frobenius. We begin with a survey of general background results on the Verschiebung, as well as certain more specialized work in fixed genus, characteristic, and rank. We then discuss t...