We identify the class of elementary groups: the smallest class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contains the profinite groups and the discrete groups, is closed under group extensions of profinite groups and discrete groups, and is closed under countable increasing unions. We show this class enjoys robust permanence properties. In particular, it is closed under group extension, taking closed subgroups, taking Hausdorff quotients, and inverse limits. A characterization of elementary groups in terms of well-founded descriptive-set-theoretic trees is then presented. We conclude with three applications. We first prove structure results for general t.d.l.c.s.c. groups. In particular, we show a compactly generated t.d.l.c.s.c. group decomposes into elementary groups and topologically characteristically simple groups via group extension. We then prove two local-to-global structure theorems: Locally solvable t.

Abstract. We show that an amenable Invariant Random Subgroup of a locally compact second countable group lives in the amenable radical. This answers a question raised in the introduction of [AGV12]. We also consider an opposite direction, property (T), and prove a similar statement for this property.

In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of chief factors for Polish groups. In the development of this theory, we prove a version of the Schreier refinement theorem. We also prove a trichotomy for the structure of topologically characteristically simple Polish groups.The development of the theory of chief factors requires two independently interesting lines of study. First we consider injective, continuous homomorphisms with dense normal image. We show such maps admit a canonical factorization via a semidirect product, and as a consequence, these maps preserve topological simplicity up to abelian error. We then define two generalizations of direct products and use these to isolate a notion of semisimplicity for Polish groups.

We first obtain finiteness properties for the collection of closed normal subgroups of a compactly generated locally compact group. Via these properties, every compactly generated locally compact group admits an essentially chief series -i.e. a finite normal series in which each factor is either compact, discrete, or a topological chief factor. A Jordan-Hölder theorem additionally holds for the 'large' factors in an essentially chief series.