Let f(x) = x(12) + ax(6) + b is an element of 7[x], with ab not equal 0. We say that f(x) is monogenic if f(x) is irreducible over Q and {1, theta, theta(2), ... , theta(11)} is a basis for the ring of integers of Q(theta), where f(theta) = 0. For each possible Galois group G of f(x) over Q, we give explicit descriptions of all monogenic trinomials f(x) having Galois group G. These results extend recent work on monogenic power-compositional quartic and sextic trinomials.
This paper deals with reductions and core of ideals in commutative rings. We examine trivial ring extensions arising from integral domains, particularly valuation and Pr & uuml;fer domains. This study builds on previous findings related to Pr & uuml;fer domains and earlier work on trivial ring extensions of special modules.
Higher-rank graphs are certain small categories that are a higher-dimensional generalisation of both directed graphs and free monoids. Path and boundary-path groupoids of finitely aligned higher-rank graphs are often constructed using either filters or graph morphisms. We generalise the graph morphism approach to finitely aligned P-graphs where (Q, P) is a weakly quasi-lattice ordered group, and we show the filter approach and the graph morphism approach yield isomorphic path and boundary-path groupoids. To do this, we define conjugacy of partial monoid actions such that conjugate actions have isomorphic semidirect product groupoids. Combining our results with others in the literature, we survey many isomorphic presentations of path and boundary-path groupoids at different levels of generality.
This paper studies non-weight representations of the Drinfeld double D of the bosonization of the Jordan plane. We give a complete classification of simple Whittaker, twisted Whittaker, and quasi-Whittaker Dmodules and determine their annihilators explicitly.
Let G be a finite group and let p be an odd prime. We denote by m(p)(G) the number of elements of order pin G. If s(p)(G) denotes the number of subgroups of order p, then m(p)(G) = (p-1)s(p)(G). Together with Frobenius' congruence s(p)(G) equivalent to 1 (mod p), this shows that the first two possible positive values of m(p)(G) are p-1 and p(2)-1. We first investigate the first threshold and obtain a solvability criterion for finite groups satisfying 0 < m(p)(G) < p(2)-1. We then study the boundary value m(p)(G) = p(2)-1. In this case, the group has exactly p + 1 subgroups of order p, and the conjugation action on these subgroups provides a natural permutation representation of degree p + 1. This action will be the main tool in the analysis of the simple case. Finally, we prove that if S is a finite nonabelian simple group and p >= 5, then mp(S) = p(2)-1 if and only if S congruent to PSL2(p). We also show that no finite nonabelian simple group satisfies m(3)(S) = 8.
