The Algebra Seminar provides graduate students, advanced undergraduate students, and faculty a forum to discuss and/or present Research, Problems, Applications, or Exposition in Algebra. For more information contact Gary Birkenmeier.
During the Fall 2017 semester we will meet on Tuesdays from 4:00-4:50 in Maxim Doucet Hall room 210.
19 September 2017
Some ideas and methods from the classification of the finite simple groups
Abstract: I intend to give a historical and high-level overview of the original program for the classification of the finite simple groups. Depending on one's point of view, this is a quest that began either in the late 1890s or in 1954, and it ended either in the early 1980s or in 2004. We will begin at the first beginning and end roughly at the first end. In the middle, the focus will be on the ideas behind the program, which partitions the collection of finite simple groups into three classes: the groups of low 2-rank (small), the groups of component type (odd), and the groups of characteristic 2-type (even). Inasmuch as there will be any details (unlikely, but this depends on one's definition of "details"), we will focus on the identification of groups in the small and odd cases.
During the Spring 2017 semester we will meet on Fridays from noon-12:50 in Maxim Doucet Hall room 214.
27 January 2017
When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice?
Abstract: (This is joint work with Martha Kilpack of Brigham Young University.)
A lattice is a partially ordered set in which every pair of elements have a least upper bound and a greatest lower bound; it can also be viewed as an algebra with two binary operators satisfying certain identities. A typical example of a lattice is the lattice of all subgroups of a given group, ordered by inclusion (or more generally, of all substructures of a given structure).
A closure operator on a partially ordered set P is a function f mapping P into P such that f satisfies three conditions:
(i) f is increasing: x less than or equal to f(x) for all x in P;
(ii) f is isotone: if x less than or equal to y, then f(x) less than or equal to f(y) for all x,y in P;
(iii) f is idempotent: f(f(x)) = f(x).
We can partially order all closure operators on P, by letting f less than or equal to g if and only if f(x) less than or equal to g(x) for all x in P. If P is a lattice, then this makes the set of all closure operators on P into a lattice.
It is a theorem of Birkhoff that every complete lattice is the lattice of subalgebras of some (possibly infinitary algebra); and a theorem of Whitman that every lattice can be embedded as a sublattice of a subgroup lattice. This leads to the question of which lattices of closure operators are isomorphic to the subgroup lattice of a group.
Previously, we had shown that if we look at the closure operators on a subgroup lattice Sub(G), then this resulting lattice, c.o.(Sub(G)) is itself a subgroup lattice if and only if G is cyclic of prime power order. We will extend the investigation first to the case of closure operators on an arbitrary finite lattice; and later, to the lattice of subgroups of an infinite group.
3 February 2017
When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 2)
10 February 2017
When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 3)
17 February 2017
When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 4)
24 February 2017
If G is an infinite group, when is the lattice of algebraic closure operators on Sub(G) isomorphic to the subgroup lattice of a group K?
3 March 2017
When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (last part)
10 March 2017 (SPECIAL TIME 12:20)
A description of indecomposable quasi-Frobenius rings
Abstract In this talk, we characterize a nilary QF-ring R in terms of the essentiality of the ideals ReR where e is a primitive idempotent of R. Recall that a ring R is nilary if AB = 0 implies either A or B is nilpotent for all ideals A, B of R. Note that nilary rings are indecomposable rings. This is a preliminary report on joint research with Omar A. Al-Mallah andHafedh M. Al-Noghashi.
24 March 2017
A description of indecomposable quasi-Frobenius rings (part 2)
31 March 2017
A description of indecomposable quasi-Frobenius rings (part 3)
7 April 2017
A description of indecomposable quasi-Frobenius rings (part 4)