Binghamton University


MATHEMATICAL SCIENCES
COLLOQUIUM


DATE: Thursday, April 8, 2010
TIME: 4:30 - 5:30 pm
PLACE: LN2205
SPEAKER: RICHARD M. FOOTE (University of Vermont)
TITLE: Strongly Closed Subgroups of Finite Groups: The Local--Global Principle in Action

Abstract


Let $G$ be a finite group with an abelian Sylow $p$-subgroup $A$, for some prime $p$. A classical result of W\. Burnside shows that if $A$ factors as $A_1 \times A_2$ under the action of the normalizer, $N_G(A)$, of $A$ in $G$, and if $N_G(A)$ acts trivially by conjugation on $A_1$, then $G$ possesses a normal subgroup $G_2$ that has $A_2$ for a Sylow $p$-subgroup. A basic (fusion) ingredient in Burnside's proof augurs the notion of {\it strongly closed subgroups} and the wealth of results on recognizing the existence of normal subgroups of a finite group from the conjugacy patterns in its $p$-subgroups. The talk will survey the history and latest results in this vein, including as a special case a complete generalization of Burnside's Theorem that provides necessary and sufficient $p$-local conditions for $G$ to factor as a direct product.

Ramifications to other areas such as algebraic topology, fusion systems, and number theory will be mentioned.


R E F R E S H M E N T S

3:30 to 4:30 PM
Kenneth W. Anderson
Memorial Reading Room