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1. CMB 2011 (vol 55 pp. 48)
Freyd's Generating Hypothesis for Groups with Periodic Cohomology Let $G$ be a finite group, and let $k$ be a field whose characteristic $p$
divides
the order of $G$.
Freyd's generating hypothesis for the stable module category of
$G$ is the statement that a map between finite-dimensional
$kG$-modules in the thick subcategory generated by $k$ factors through a
projective if the induced map on Tate cohomology is trivial. We show that if
$G$
has periodic cohomology, then the generating hypothesis holds if and only if
the Sylow
$p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions
that are equivalent to the GH
for groups with periodic cohomology.
Keywords:Tate cohomology, generating hypothesis, stable module category, ghost map, principal block, thick subcategory, periodic cohomology Categories:20C20, 20J06, 55P42 |
2. CMB 2011 (vol 55 pp. 38)
Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras
Zarhin proves that if $C$ is the curve $y^2=f(x)$ where
$\textrm{Gal}_{\mathbb{Q}}(f(x))=S_n$ or $A_n$, then
${\textrm{End}}_{\overline{\mathbb{Q}}}(J)=\mathbb{Z}$. In seeking to examine his
result in the genus $g=2$ case supposing other Galois groups, we
calculate
$\textrm{End}_{\overline{\mathbb{Q}}}(J)\otimes_{\mathbb{Z}} \mathbb{F}_2$
for a genus $2$ curve where $f(x)$ is irreducible.
In particular, we show that unless the Galois group is $S_5$ or
$A_5$, the Galois group does not determine ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)$.
Categories:11G10, 20C20 |
3. CMB 2006 (vol 49 pp. 285)
Orbits and Stabilizers for Solvable Linear Groups We extend a result of Noritzsch,
which describes the orbit sizes in the action of a
Frobenius group $G$ on a finite vector space $V$ under
certain conditions, to a more general class of finite
solvable groups $G$.
This result has applications in computing
irreducible character degrees of finite groups.
Another application, proved here, is a result
concerning the structure of certain groups with
few complex irreducible character degrees.
Categories:20B99, 20C15, 20C20 |
4. CMB 2006 (vol 49 pp. 96)
Roots of Simple Modules We introduce roots of indecomposable modules over group algebras of finite groups,
and we investigate some of their properties. This allows us to correct an error
in Landrock's book which has to do with roots of simple modules.
Categories:20C20, 20C05 |
5. CMB 2000 (vol 43 pp. 79)
Cyclotomic Schur Algebras and Blocks of Cyclic Defect An explicit classification is given of blocks of cyclic defect of
cyclotomic Schur algebras and of cyclotomic Hecke algebras, over
discrete valuation rings.
Categories:20G05, 20C20, 16G30, 17B37, 57M25 |