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The Algebra Seminar
Fall 2011
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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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August 30: Organizational meeting
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September 6 : Adam Perry
Title: Information learned from the diagonal of a direct product
GxG in finite groups.
Abstract: Let A,B and G be finite groups. Goursat's theorem
tells us for
U a subgroup of AxB that the projections of U onto the direct factors mod the
intersections of U with the direct factors must be isomorphic and that the
image of U is a diagonal. With that being said, we will discuss what can
be understood about G if the diagonal is related to GxG in a given way.
Specifically, the diagonal of GxG is normal(or subnormal) if and only if G
is abelian(or nilpotent resp.). After discussing these cases, we will
show that diagonal is modular in GxG if and only if G is abelian. If time
permits, we will see what happens if the diagonal is submodular in GxG.
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September 15 (SPECIAL DAY AND TIME: Thursday 2:50): Luise-Charlotte
Kappe
Title: On the classification of special p-groups of rank 2 that
appear as central quotient groups.
Abstract: We say that a group is capable if it appears as a central quotient group. A
group is a special p-group of rank 2 if its commutator subgroup is equal to
its center and is an elementary abelian p-group of rank 2. According to a
1990 result by H. Heineken, special p-groups of rank 2 which are capable
have order p^5, p^6, or p^7. For odd p, we classify special p-groups of
rank 2 and order p^5 which are capable and give an outline for the cases
when the orders are p^6 or p^7. In conclusion we will make some remarks on
what is known for p = 2.
This is joint work with H.Heineken and R.F.Morse.
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September 20: Marcos Zyman (CUNY)
Title: IA-automorphisms of groups with almost constant upper central series
Abstract: Let G be any group for which there exists a positive integer j
such that Z_j=Z_{j+1} in the upper central series. Let IA(G) be the subgroup
of Aut(G) consisting of those automorphisms that induce the identity on G
abelianized. Define the group of j-central automorphisms as those elements
of Aut(G) that induce the identity on G/Z_j.
I will offer sufficient conditions for IA(G) to have a useful direct product
structure involving j-central automorphisms, and apply these results to
the IA-group of certain finitely generated center-by-metabelian groups.
I also plan to provide some background results on IA-automorphisms of
nilpotent and solvable groups.
This is joint work with M. Bonanome and M. H. Dean.
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September 27: Eric Swartz
Title: The locally 2-arc transitive graphs admitting an almost
simple group of Suzuki type
Abstract:
Abtsract
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October 4: Elizabeth Wilcox (Colgate University)
Title:
Sizing up subgroups of finite groups
Abstract: In this talk I'll define the Chermak-Delgado measure of a subgroup H of a
finite group G and show that all subgroups with maximal measure form a
sublattice within the subnormal subgroup lattice of G. I'll discuss some
recent results from an ongoing collaboration between Ben Brewster and myself
and, as a consequence, I'll most definitely talk about direct products and
wreath products. I will review the definition of a wreath product before
digging in for a few calculations and give some motivation for our current
investigations.
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October 11: Marcin Mazur
Title: Linear algebra and finite p-groups
Abstract: In this talk I will show how questions about special p-groups
are related to some interesting problems in linear algebra, in particular
answering some problems posed in Luise-Charlotte Kappe talk earlier this
semester. This is a joint project with H.Heineken, L-C Kappe, and R.F.Morse
and should be considered as a follow up of the talk by L-C Kappe.
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October 13 (CROSS LISTING WITH GEOMETRY AND TOPOLOGY SEMINAR; SPECIAL DAY:
Thursday, 2:50 p.m.) : Peter Kropholler (Glasgow and Cornell)
Title: Wilson's short proof of the Romanovskii--Wilson Theorem
Abstract:
The theorem says this: let $m$ and $n$ be natural numbers with $n> m$.
Suppose you have a group $G$ which admits a presentation with $n$
generators
and $m$ relators. Then for any set $Y$ of generators of $G$, there is a
subset of
$n-m$ elements of $Y$ that freely generate a free group of rank $n--m$.
It is proved by using ordered groups and embeddings in division rings to
reduce it to
the following statement about finite dimensional vector spaces: if $V$ is
an $n$
dimensional vector space and $U$ is an $m$-dimensional subspace then any
subset $Y$ of
of $V$ which spans $V$ modulo $U$ contains a subset of $n-m$ vectors which
span a
complement to $U$ in $V$.
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October 18: Dikran Karagueuzian
Title: Betti numbers of resolutions and free actions of groups on products of spheres
Abstract: A conjecture in commutative algebra asserts that the Kozsul resolution
is minimal, at least in the sense that no other resolution has smaller
Betti numbers. There is an analogous topological conjecture: if an
elementary abelian group acts freely on a product of spheres, the
number of spheres is at least as large as the rank of the group.
These conjectures were unified by a theorem of Gunnar Carlsson, which
shows that they are both special cases of a conjecture on the rank of
differential graded modules.
This talk will be an introduction to all three conjectures and a
discussion of attempts to make progress on the differential graded
modules version.
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October 25: David Biddle
Title: On the Size and Connectivity of Graphs of Generating
Sets of Finitely Generated Groups.
Abstract:
Abtsract
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November 1: No talk this week
Title:
Abstract:
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November 8: David Biddle
Title: On the Size and Connectivity of Graphs of Generating
Sets of Finitely Generated Groups, part II.
Abstract:
We will analyze the method(s) needed to prove the connectivity
condition(s) of the PR graphs of finite abelian groups and we will be
looking at extensions of these results to larger classes of groups. We
will also discuss sharper bounds for S_n and A_n for 4 < n.
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November 15: Diego Penta
Title: On the exponential map from a Lie algebra to a Lie group
Abstract: We will explore the connection between Lie algebras and Lie
groups by defining the exponential map, and investigating some of its
properties.
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November 22: Michael Hampton
Title: Sheaf Spaces and Sheaves of Universal Algebras
Abstract: This will be an expository talk on the basics of sheaf spaces
and sheaves of universal algebras with a mention of their role in natural
dualities.
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November 29: Craig DeFelice
Title:
Counting subrings of Z^n of index k (after R. I. Liu).
Abstract: We investigate the number of subrings of the ring
Z^n of a fixed index k (denoted f_n(k)). We present a way of decomposing
subrings to help
in counting. And lastly will calculate f_n(p^e) when e<6. The talk is based
on a paper by R. I. Liu.
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December 6: Ryan McCulloch
Title: On the Nilpotence of Frobenius Kernels (Part I of
two talks for Candidacy Exam.)
Abstract: In these talks we present Thompson's Ph.D. thesis that Frobenius kernels
are nilpotent. The strength of the theorem is Thompson's normal
p-complement theorem, which can be viewed as a strong form of Frobenius's
normal p-complement theorem. We hope to get through as much of the
details as possible in these two talks, providing necessary background
information when needed.
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