|Prof. Mark Lewis, Ph.D.
Kent State University USA
Groups where the centers of the irreducible characters form a chain
Throughout our talk, all groups will be finite. We consider groups where the centers of the irreducible characters form a chain. We obtain two alternate characterizations of these groups. One of these characterizations involves the centers of all quotients of the group. The other characterization looks at finding a chain of normal subgroups that have a given problem. We obtain some information regarding the structure of these groups. In particular, we obtain a necessary and sufficient condition for a nested group to be nilpotent. Using our results, we are able to classify those groups where the kernels of the irreducible characters form a chain. We also classify the groups where the kernels of the nonlinear irreducible characters form a chain. This generalizes a result of Qian and Wang that answered a question that was posed by Berkovich.
A GVZ-group is a group where every irreducible character vanishes off its center. We consider some alternate definitions of nested groups, and we show for GVZ-groups, that these different definitions are equivalent. We present examples that show that these different definitions are not equivalent for all groups. We show that a result of Nenciu regarding nested GVZ groups is really a result about nested groups. We give strong results regarding nested groups that are nilpotent of nilpotence class 2. Finally, we obtain an alternate proof of a theorem of Isaacs regarding the existence of p-groups with a given set of irreducible character degrees.
|Prof. Mahmut Kuzucuoglu, Ph.D.
Middle East Technical University Turkey
Limit Monomial Groups
There are three principal types of representations of groups. These are (1) Permutation representation (2) Linear representation (3) Monomial representation. The permutation representation and linear representation of groups are studied extensively and the main properties of these representations are well known.
The finite degree monomial representations of groups are studied by O. Ore in . The infinite degree monomial representations are studied by R. B. Crouch in . The basic properties of direct limit of monomial groups of finite degree is studied in . The structure of centralizers of elements in limit monomial groups, the classification of such groups using Steinitz numbers and isomorphisms of limit monomial groups are studied in . We mention the following theorem.
Theorem. Let λ and μ be two Steinitz numbers. The homogeneous monomial groups Σλ(H) and Σμ(G) are isomorphic if and only if λ=μ and H≅G provided that the splittings of Σλ(H) and Σμ(G) are regular.
We plan to discuss, construction of direct limit of infinite degree monomial groups over arbitrary group H. Then the basic properties of constructed limit group, the structure of centralizers of elements and conjugacy of two elements in limit monomial groups will be studied.