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Analogs of Clifford's Theorem for Polycyclic-by-finite Groups
| Content Provider | Semantic Scholar |
|---|---|
| Author | Lorenz, Martin |
| Copyright Year | 2008 |
| Abstract | Let P be a primitive ideal in the group algebra K[G] of the polycyclic group G and let N be a normal subgroup of G. We show that among the irreducible right K[G]-modules with annihilator P there exists at least one, V, such that the restricted K[N]-module VN is completely reducible, a sum of G-conjugate simple K[N]-submodules. Various stronger versions of this result are obtained. We also consider the action of G on the factor K[N]/P n K[N] and show that, in case K is uncountable, any ideal I of K[N] satisfying ngfl gI= P n K[NJ is contained in a primitive ideal Q of K[N] with n gEGQ8 = P n K[N]. Introduction. By the classical restriction theorem of Clifford, we know that, given a finite group G and an irreducible right K[G]-module V, then V restricted to any normal subgroup N of G is a completely reducible K[N]module and its simple components are conjugate under G. In the case of infinite groups, this result is still true provided the normal subgroup N has finite index in G (cf. [15, Theorem 7.2.16]), but it fails to hold in general without this restriction on N. So consider, for instance, the two-generator nilpotent group of class two, G = . If K is a field containing an element X E K of infinite multiplicative order, then the Laurent polynomial ring V = K[X + 1] becomes an irreducible right K[G]-module by the rules X' x = X'+I, xi y = X'iX, Xi. z = AX'. But for the normal subgroup N = of G, the restricted module VN has no irreducible K[N]-submodule, because such a submodule would correspond to a minimal ideal in K[X ? 1]. However, there is still something that can be said for an arbitrary normal subgroup N of the polycyclic-by-finite group G. Namely, let P be a primitive ideal in K[G], say P is the annihilator of the irreducible right K[G]-module V. In case VN splits into a sum of G-conjugate simple K[N]-submodules, an immediate consequence would be that |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://math.temple.edu/~lorenz/papers/6.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Amoxicillin Contain (action) Immunostimulating conjugate (antigen) Irreducibility Laurent polynomial Microsoft Dynamics AX Polynomial ring Rule (guideline) Subgroup A Nepoviruses Version |
| Content Type | Text |
| Resource Type | Article |
