# Chief factors and chief series in locally compact groups

#### Philip Wesolek, Binghamton University. Part of the pure mathematics algebra seminar series.

(Joint work with C. Reid.) In locally compact groups, studying the tension between topological structure and geometric structure often yields surprising, general results. In this talk, we show the normal subgroup structure of a  locally compact group is restricted by this tension. A closed normal factor $K/L$ of a locally compact group $G$ is called a chief factor if there is no closed normal subgroup of $G$ strictly between $L$ and $K$. We show that every compactly generated locally compact group $G$ admits a finite series $\{1\}=G_0\leq G_1\leq \dots \leq G_n=G$ of closed normal subgroups so that each normal factor $G_i/G_{i-1}$ is either discrete, compact, or chief; such a series is called an essentially chief series. We then demonstrate a uniqueness result for essentially chief series. Using the existence and uniqueness of essential chief series for compactly generated groups, we go on to prove that any sufficiently complex locally compact group admits chief factors; i.e. we can relax the compact generation hypothesis. As a corollary, general structure results are obtained for chief factors.

After each talk at 4:15pm there will be tea/coffee in the Common Room at level 9 in Maths building. The current seminar organiser is Eleonore Faber.

Philip Wesolek, Binghamton University