-covering systems of subgroups for classes of -supersoluble and -nilpotent finite groups.
We describe a correspondence between -invariant tensors and graphs. We then show how this correspondence accommodates various types of symmetries and orientations.
Suppose is a commutative unital ring and is an abelian group. We give a general criterion only in terms of and when all normalized units in the commutative group ring are -nilpotent. This extends recent results published in [Extracta Math., 2008–2009] and [Ann. Sci. Math. Québec, 2009].
For a prime number l and for a finite Galois l-extension of function fields L / K over an algebraically closed field of characteristic p <> l, it is obtained the Galois module structure of the generalized Jacobian associated to L, l and the ramified prime divisors. In the cyclic case an implicit integral representation of the Jacobian is obtained and this representation is compared with the explicit representation.
For , any totally ramified cyclic extension of degree of local fields which are finite extensions of the field of -adic numbers, we describe the -module structure of each fractional ideal of explicitly in terms of the indecomposable -modules classified by Heller and Reiner. The exponents are determined only by the invariants of ramification.
The main results of this paper may be loosely stated as follows.Theorem.— Let and be sums of Galois algebras with group over algebraic number fields. Suppose that and have the same dimension and that they are identical at their wildly ramified primes. Then (writing for the maximal order in )In many cases The role played by the root numbers of and at the symplectic characters of in determining the relationship between the -modules and is described. The theorem includes...