| 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 |
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.