Let $$G$$ be a finite group. Its spectrum $$\omega(G)$$ is the set of all element orders . The prime $$\pi(G)$$ divisors order Gruenberg–Kegel graph (or graph) $$\Gamma(G)$$ simple with vertex in which any two vertices $$p$$ and $$q$$ are adjacent if only $$pq\in\omega(G)$$ structural theorem implies that class groups disconnected graphs widely generalizes Frobenius groups, whose role group the...

A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, in...

The representation theory of finite groups began with Frobenius's factorization Dedekind's group determinant. In this paper, we consider the case semigroup determinant is nonzero if and only complex algebra Frobenius, so our results include applications to study Frobenius algebras. We explicitly factor for commutative semigroups inverse semigroups. recover Wilf-Lindström a meet semilattice Wood...

lsmr (least squares minimal residual) is an iterative method for the solution of the linear system of equations and leastsquares problems. this paper presents a block version of the lsmr algorithm for solving linear systems with multiple right-hand sides. the new algorithm is based on the block bidiagonalization and derived by minimizing the frobenius norm of the resid ual matrix of normal equa...

Class field theory relates abelian extensions of a given number field K to certain generalized ideal class groups ofK. The fundamental tool for doing this is Frobenius elements, and the Artin map obtained from them. Recall from lecture 24 that if L/K is an abelian extension (i.e., a Galois extension with abelian Galois group), and p is a prime of OK unramified in OL, then we defined Fr(p) ∈ Gal...

Techniques and results of constructing balanced incomplete block designs from a Frobenius group were summarized in [2]. These techniques and results grew out of observing that the sets Na + b of a planar near-ring (N, +, .) usually have a geometric interpretation. This observation was first illustrated in [1]. In this same work, one notices that sometimes the sets N{a,-a} + b also have geometri...

The G-function associated to the semisimple Frobenius manifold C/W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the Gfunction is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the G-function corresponding to the Frobenius manifold C/W̃ (An−1) , where W̃ (An−1) is a...

Let G be a simple algebraic group defined over a finite field of good characteristic, with associated Frobenius endomorphism F. In this article we extend an observation of Lusztig, (which gives a numerical relationship between an ordinary character of GF and its unipotent support), to the case where Z(G) is disconnected. We then use this observation in some applications to the ordinary characte...