The Algebra Seminar
Fal 2006
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
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August 29: Organizational meeting
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September 5: Marcin Mazur
Title: First cohomology of the Klein's four-group.
Abstract:
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September 12: Peter Hauck (Eberhard-Karls-Universitat Tubingen)
Title: 2-generated subgroups of finite groups.
Abstract: Let L be a class of finite groups. Two subgroups A, B
of a group G are called L-connected if < a,b > is in L for all a in A and b in B.
This concept was introduced by A. Carocca in 1966 and has received considerable interest recently (for instance in a paper of J. Beidleman and H. Heineken from 2004).
The leading question for the talk is the following: What can be said about a finite group that is generated by two L-connected subgroups? We present some
answers to this question for various important classes L lying between
the class of all finite abelian groups and the class of all finite solvable
groups of nilpotent length at most 2.
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September 19: John Loftus
Title: Context-free languages seen from the perspective of powers of primitive words
Abstract: I'll introduce a new class of languages defined in terms of
powers of primitive words, and prove that you can't decide if a language
defined by a context-free grammar is in the new class. This is based on
work previously done with respect to regular languages.
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September 26: John Loftus
Title: Bounded languages and powers of primitive words
Abstract: I'll summarize some results from Horvath and Ito(2001)
describing relations between a language and its support, then outline
the proof for an analogous result based on boundedness for languages.
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October 3: Gabriela Mendoza
Title: For a given prime p, what is the smallest nonabelian simple group
whose order is divisible by p?
Abstract: In this talk we will show that for any prime p>3, the group
PSL(2,p), the Projective Special Linear group of rank 2 over the field of p
elements and which has order p(p^2-1)/2, is the smallest nonabelian simple
group whose order is divisible by p. For p=2 and 3 the group in question is
A_5, the alternating group on 5 letters.
This answers a question posed at the Zassenhaus Conference 2005 at the
University of Alabama in Montgomery.
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October 10 : Joseph Evan
Title: Subgroups that Satisfy the Strong Frattini Argument
Abstract: A subgroup U of a group G satisfies the Frattini Argument in
G if for all normal subgroups N of G, the product of N with the
normalizer of the intersection of U and N is equal to all of G. A
subgroup then satisfies the strong Frattini Argument in G if it
satisfies the Frattini Argument in all subgroups of G in which it is
contained. In this talk we will examine some properties of subgroups
that satisfy the strong Frattini Argument and subgroups of direct
products that satisfy the strong Frattini Argument.
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October 17: Arny Feldman, Franklin & Marshall University
Title: Injectors in Finite Solvable Groups
Abstract: Abstract
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October 24: Alex Feingold, Binghamton University
Title: Cartan Matrices, Dynkin Diagrams, and Root Systems
Abstract: I will give an introduction to the theory of root systems,
and how they are encoded in either a Cartan matrix or a Dynkin diagram. These are
usually used for the classification of finite dimensional simple Lie algebras over
the complex numbers, but I will avoid discussing Lie theory in this talk. Basic
definitions and explicit examples will be emphasized, but if time allows, I will
also discuss generalizations of these concepts from the finite types to the
infinite types (affine and indefinite) which appear in the theory of infinite
dimensional Kac-Moody Lie algebras.
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October 31: Ilir Snopce
Title: Introduction to Nottingham groups.
Abstract: The Nottingham group N=N(F), where F is a finite field
of characteristic p, is the group of formal power series t+a_1t^2+a_2t^3+... over F under
composition
or, equivalently, the group of automorphisms of the ring F[[t]] which act
trivially on tF[[t]]/t^2F[[t]].
This finitely generated pro-p group is not soluble, is of finite width
(that is, the succesive quotients of
the lower central series have bounded order),
and is just infinite as an abstract group (that is, any non-trivial normal
subgroup is of finite
index). The most interesting property of the Nottingham group is that every
countably based pro-p
group can be embedded into N as a closed subgroup. These properties make
the Nottingham group
a favorite test case for conjectures concerning pro-p groups.
I will start by giving an introduction to Nottingham groups. Then I will prove
a theorem as a consequence of which we will get that the Nottingham groups are
just infinite as abstract groups.
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November 7: Elizabeth Wilcox
Title: Subnormal subgroups and finite
solvable T-groups.
Abstract: We will begin with the definition of a subnormal subgroup, as
well as some examples, and ultimately show that in a finite group the
collection of subnormal subgroups forms a complete sublattice of the
lattice of subgroups. We will then work with T-groups, groups in which
subnormal subgroups are actually normal, building towards a
characterization of all finite solvable T-groups.
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November 14: Darryl Daugherty
Title: Packings, free products and residually finite groups.
Abstract I will discuss a paper by A.M. Macbeath,
"Packings, free products and residually finite groups.",
Proc. Cambridge Philos. Soc. 59 1963 555--558.
From Math. Reviews:
"In this note a simple principle is explained for constructing a
transformation group which is a free product of given transformation groups.
The formulation is purely settheoretic, without any topology, and it can
apply to any free product, whatever the cardinal number of the set of
factors. The principle is used to establish the closure under the formation
of a countable free product of the family of groups which can be represented
as "discontinuous" subgroups of a certain group of rational projective
transformations. These results are then applied to give a new proof of the
theorem of Gruenberg [Proc. London Math. Soc. (3) 7 (1957), 29--62;
MR0087652 (19,386a)] that a free product of residually finite groups is
itself residually finite."
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November 21: Nor M. Mohd Ali, University of Technology of Malaysia
Title: On The Capability of Groups of Nilpotency Class Two
Abstract: A group is called capable if it is a central factor group. R.Baer
characterized finitely generated abelian groups which are capable as
those groups which have two or more factors of maximal order in their
direct decomposition. Using the explicit knowledge of the nonabelian
tensor square of 2-generator p-groups of nilpotency class two, p odd,
M.Bacon and L.C.Kappe characterized the capable ones among those groups.
The topic of this talk is to characterize the capable groups in the
class of infinite 2-generator groups of nilpotency class two. The
methods are similar to those used by Bacon and Kappe.
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November 28 : Eric Sponza
Title: Rottlander Groups
Abstract: Two groups are said to have identical subgroup structure if there
exists an index and conjugacy preserving projectivity between them.
In this talk, I'll present the structure of the Rottlander groups and
show that they give us examples of nonisomorphic groups of the same
order with identical subgroup structure.
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December 5: Dandrielle Lewis
Title: Preparation for a group theoretic proof of Burnside's
p^aq^b theorem.
Abstract:
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