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Modules over group algebras which are free on restriction to a maximal subgroup
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Andrews, Robert Charles (1987) Modules over group algebras which are free on restriction to a maximal subgroup. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3141941~S15
Abstract
Consider the following situation: k will be an algebraically closed field of characteristic p and G will be a finite p-group, V will be a non-projective, indecomposable kG-module which is free on restriction to some maximal subgroup of G. Our purpose in doing this is to investigate Chouinard's theorem - all the proofs of which have been cohomological in nature - in a representation-theoretic way. This theorem may be shown to be equivalent to saying that, if G is not elementary abelian, V cannot be free on restriction to all the maximal subgroups of G.
It is shown how to construct an exact sequence:
O → V → P → P → V → O
with P projective. From this an almost split sequence,
O → V → X → V → O
is constructed. It is shown that X can have at most two indecomposable summands.
If φ denotes the Frattini subgroup of G, then V is free on restriction to φ. We may regard the set of φ-fixed points of V, V̄, as a module for Ḡ =G/φ. But Ḡ is elementary abelian, so we may consider the Carlson variety, Y(V̄) - this may be regarded as a subset of J/J² where J denotes the augmentation ideal of kG. It is shown that Y(V̄) is always a line.
We define YG to be the union of all the lines Y(V̄) as V runs over all the kG-modules with the properties above. It is shown that YG is the whole of J/J² if and only if G is elementary abelian. It is also shown that, when G is one of a particular class of p-groups - the pseudo-special groups - which form the minimal counter-examples to Chouinard's theorem, that YG is the set of zeros of a sequence of homogeneous polynomials with coefficients in the field of p elements. Indeed, a specific construction for these polynomials is given.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Group algebras, Modules (Algebra), Maximal subgroups, Abelian groups | ||||
Official Date: | August 1987 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Green, J. A. (James Alexander) | ||||
Sponsors: | Science and Engineering Research Council (Great Britain) | ||||
Format of File: | |||||
Extent: | xiii, 200 leaves : illustrations | ||||
Language: | eng |
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