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The Algebra Seminar
Fall 2012
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The seminar meets Tuesdays in room LN 2205 at 2:50 p.m. There will be refreshments in the Anderson Reading Room at 4:00.
Organizers: Alex Feingold and Marcin Mazur
To receive announcements of seminar talks by email, please join the seminar's
mailing list.
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September 11 : Ben Brewster
Title: Some recent results concerning the Chermak-Delgado lattice
in finite p-groups
Abstract: If H is a subgroup of group G, mG(H)=|H| |C(H)|,
where C(H) is the centralizer in G of H. Let m*(G)= max{mG(H) |H a subgroup of
G}. Let *CD(G) *= {H | mG(H)=m*(G) }.
It is known that *CD(G) *is a self-dual modular sublattice of the
lattice of subnormal subgroups of G and that any two elements of *CD(G)
*permute.
In this talk I will explain some details involved in verifying the
comments above and then demonstrate one approach to constructing a group
G with a prescribed self dual modular lattice as *CD(G)*. I will focus
on the lattices which are totally ordered and,hopefully,give a glimpse
of how one can see that every finite chain is *CD(P)* for p an arbitrary
prime and P a p-group of class 2.
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September 18: No meeting (Rosh Hashanah)
Title:
Abstract:
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September 25: No meeting (Yom Kippur)
Title:
Abstract:
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October 2: Ryan McCulloch
Title: Consequences of a Minimal Maximal Chermak-Delgado Measure
Abstract:
As defined in the first talk this Fall, given a finite group G
and a subgroup, H, of G, m_G(H) = |H||C_G(H)| is the Chermak-Delgado
measure of H,
m*(G)= max{m_G(H) | H is a subgroup of G}, and CD(G) = {H |
m_G(H)=m*(G)} is the Chermak-Delgado lattice of G.
It is easy to show that for any group G, m*(G) is between |G| and |G|^2.
Groups which attain the upper bound on this inequality are precisely the abelian
groups, and their Chermak-Delgado lattices consist of a single point which
is the group itself. The aim of this talk will be to examine when m*(G) =
|G|. Some negative consequences are that such a group G cannot factorize
nontrivially into two abelian subgroups, that no normal nontrivial cyclic
groups can be in CD(G), that no nontrivial p-groups can be in CD(G), and
that no normal subgroups which have one "very large" prime divisor can be
in CD(G). We do have an example where CD(G) is a three point chain.
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October 9: Alex Feingold
Title: Introduction to Vertex Operators and Representations of Infinite
Dimensional Lie Algebras
Abstract: The infinite dimensional Kac-Moody Lie algebras and related
Virasoro algebra have played
an important role in mathematics and physics. Representations of these
algebras can be constructed
using ``vertex operators" used by physicists in string theory and
conformal field theory. This talk
will be an introduction to this subject starting with a brief introduction to
finite dimensional Lie algebras.
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October 16: Amanda Taylor
Title: Locally Solvable Subgroups of PLo(I)
Abstract: A locally solvable group is a group in which every finitely generated subgroup
is solvable. We introduce a geometric criterion that is equivalent to local solvability in PLo(I) and
discuss a proof that locally solvable subgroups of PLo(I) are countable. Classification of these
subgroups is the subject of my thesis. All these results hold for Thompson's Group F, too.
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October 23: Conchita Martínez-Pérez (Universidad de Zaragoza)
Title: Isomorphisms of Brin-Higman-Thompson groups
Abstract: This is a joint work with Warren Dicks. We prove that the
Brin-Higman-Thompson groups
$sV_{r,n}$ are isomorphic to certain groups of matrices over Leavitt
algebras. This was
observed by Pardo in the case $s=1$. Then using arguments available in
the literature we
completely determine the isomorphisms classes between the groups
$tV_{r,n}$ (the case
$t=1$ was also obtained by Pardo).
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October 30: No meeting
Title:
Abstract:
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November 6: David Biddle
Title: Orbit structures on the Generating Tuples of a Finite Group
Abstract: Let G be a finite group and look at the collection
S_k of generating k-tuples of G. S_k admits 3 related actions: 1. An action of
Aut(F_k) [Nielsen operations] 2. The diagonal action from Aut(G), and the
product action 3. Aut(F_k) x Aut(G) [T_k-systems]. For large classes of groups
and suitable k, the orbit structure from these actions is well understood
(for instance, f.g. abelian groups). In this talk we will describe ways of
distinguishing orbits of actions #1 and #3 using determinant and torsion
invariants and give a characterisation of what groups have precisely 1 orbit
under action #2. Recent results and open problems will be discussed.
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November 13: Diego Penta
Title: Exploring hyperbolic Kac-Moody Lie algebras using Dynkin diagrams
Abstract: The theory of hyperbolic Kac-Moody algebras naturally
generalizes the theory of affine Kac-Moody algebras, which is itself
the most natural generalization to infinite dimensions of finite
dimensional Lie theory. In this talk I will give some background into
K-M theory, and investigate the classification of the hyperbolic K-M
algebras based on their associated Dynkin diagrams. I will show how a
Dynkin diagram can be used to determine whether the associated K-M
algebra is symmetrizable. I will also show that the maximal rank of a
K-M algebra of compact hyperbolic type is 5, while the maximal rank of
a symmetrizable K-M algebra of compact hyperbolic type is 4.
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November 20: Andrew Linshaw
Title: Chiral equivariant cohomology
Abstract: We describe a new equivariant cohomology that takes values in a
vertex algebra, and contains and generalizes the classical equivariant cohomology of a
manifold with a Lie group action. This cohomology has interesting features with no
classical analogues, such as a conformal structure when the group is semisimple.
For any simple group G, there is a sphere with infinitely many actions of G with the same
classical equivariant cohomology, which can all be distinguished by this new invariant.
(Joint work with B. Lian and B. Song).
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November 27: Rachel Skipper
Title: An upper bound on the Minimum Number of Generators of a Finite p-Group
Abstract: Thomas J. Laffey's argument shows d(S), the minimum number of
generators of a p-group, S, p>2, is less than or equal to \log_p |
\omega_1 (S)|. The result was shown by J.G. Thompson for p-groups for
which all elements of order p are central, but Laffey gives the result in
more generality. The major theorem that leads up to this result proves the
existence of a normal subgroup, K, satisfying certain properties, namely
d(S)=d(K), \Phi(K)=K', K' \leq Z(K), and K' is elementary abelian. The
proof of this theorem also adopts an argument from Thompson's ``A
replacement theorem for p-groups and a conjecture."
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December 4: Michael Hampton
Title: Boolean constructions in universal algebra
Abstract: Boolean products are a special sheaf construction used in
universal algebra which can yield insight into algebraic structures. In
this talk, we will review the basic notions from universal algebra,
introduce Boolean products, and present an Ershov translation for filtered
Boolean powers. Time permitting, we will indicate the role this plays in a
decidability result for certain classes of Boolean products.
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December 11: Joseph Brennan
Title: Finite non-abelian p-groups possessing an abelian subgroup of index p
Abstract: The classification of finite p-groups possessing a cyclic subgroup of index
p appears in Burnside's "Theory of Groups of Finite Order" (1897) and the
results will be familiar to many readers. My current research, and the
subject of my talk, is the structure of non-abelian finite p-groups
possessing an abelian subgroup of index p. I will be reporting my current
progress and presenting the classification of non-abelian finite p-groups
possessing an abelian subgroup of index p and a cyclic center.
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