‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $f$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2‎, ‎3‎, ‎7)$-generated and of those which are $(2,3)$-generated‎.

‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $mathbb{f}$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2,3,7)$-generated and of those which are $(2,3)$-generated‎.

Abstract A Hurwitz generating triple for a group

We consider a mathematical model of epidemic spread  in which the  population  is partitioned  into five compartments of susceptible S(t), Infected I(t), Removed R(t), Prevented U(t) and the Controlled W(t). We assume each of the compartments comprises of cohorts of individuals which are  identical with respect to the disease status. We derive five systems of equations to represent each of the ...

Hurwitz numbers count ramified genus g, degree d coverings of the projective line with fixed branch locus and fixed ramification data. Double Hurwitz numbers count such covers, where we fix two special profiles over 0 and ∞ and only simple ramification else. These objects feature interesting structural behaviour and connections to geometry. In this paper, we introduce the notion of pruned doubl...

This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of part...

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of the branched covers counted by the Hurwitz numbers, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzyks...

Let , 3, 7) denote the infinite triangle group, defined by the presentation X,Y | X = Y 3 = (XY ) = 1 . A non-trivial group G is said to be (2, 3, 7) -generated (or a Hurwitz group, when finite) if it is an epimorphic image of , 3, 7). Hurwitz groups are particularly interesting for the theory of Riemann surfaces. Namely, if H is the automorphism group of a compact Riemann surface of genus g > ...