This page is best viewed using the open source browsers
Mozilla or
Mozilla Firebird.
Besides supporting math fonts that Internet Explorer does not,
these are generally faster at rendering (except for some graphics), more
secure, can block unwanted popups, and give explicit control over cookies.
Jennings' theorem for p-split groups, submitted to the Journal
of Algebra.
Darren Semmen
Website: http://math.uci.edu/~dsemmen
Institution: University of California, Irvine
Abstract:
Jennings [7] used the descending central series of dimension
subgroups to produce a basis of the radical layers of the group ring of
a p-group over a field k of
characteristic p and, with it, a Hilbert polynomial for the
dimensions
of these layers.
Later, Quillen [9] employed Lie ring methods of
Lazard [8] to refine the proof of
Jennings' theorem. Analogously, Alperin [1] extended
Jennings' methods to
permutation modules for p-groups, while Shalev [12] later
recast this in terms of Quillen's formulation.
All of this naturally extends to p-split groups,
finite groups with normal p-Sylow.
Consider the filtration of the group ring kG by the
powers of its Jacobson radical. When G is a p-group,
Quillen
showed that the associated graded ring is isomorphic to the graded
p-restricted universal enveloping algebra
UJ*⊗k(G)
for the
graded p-restricted
Lie algebra generated from the filtration of G by its descending
central sequence of dimension subgroups. For a p-split group
G
with p-Sylow P and a given maximal subgroup A of
order
prime to p,
the associated graded ring is isomorphic to the skew group ring
UJ*⊗k(P)xsA
(Theorem 4),
where A is concentrated in degree zero and the
action of A on
UJ*⊗k(P)
is induced from its conjugation action on P.
Let H be a subgroup of G and S a semisimple
kH-module. Consider the graded module associated to the
filtration
by the powers of the Jacobson radical of the module formed by inducing
S
up to PH. This graded module will be isomorphic to the Hopf
algebra
tensor product of S (regarded as a semisimple kPH-module)
with the quotient of
UJ*⊗k(P)
by a left ideal generated by
elements
corresponding to q-1, where the q are elements of the
p-Sylow of
H; inducing this graded module up to G will preserve the
grading (Theorem 5).
Hilbert polynomials for the Brauer character of the radical layers
allow explicit computation of the isomorphism classes of the
radical layers of the module produced by inducing S to G.