(Joint work with Colin Reid.) Compactly generated locally compact groups appear throughout mathematics. Connected locally compact groups are examples, and such groups admit a detailed structure theory via the celebrated solution to Hilbert's fifth problem and Lie theory. However, many - one could argue most - compactly generated locally compact groups are totally disconnected! Furthermore, a growing body of results show that totally disconnected locally compact (t.d.l.c.) groups also admit a rich, tractable theory.

A fundamental question concerning compactly generated t.d.l.c. groups is whether or not such a group can the be `decomposed' into `basic' pieces; in the connected case, the answer is `yes' via the solution to Hilbert's fifth and Lie theory. The answer in the compactly generated t.d.l.c. case is also `yes'! A closed normal factor $K/L$ of a t.d.l.c. group $G$ is called chief if there is no closed normal subgroup of $G$ strictly between $L$ and $K$. We show that every compactly generated t.d.l.c. 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, profinite, or chief; such a series is called an essentially chief series. We go on to demonstrate a uniqueness result for essentially chief series analogous to the Schreier refinement theorem in finite group theory. Finally, a trichotomy theorem for chief factors is noted.