  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.d.l.c.s.c.
groups are elementary and [A]-regular t.d.l.c.s.c. groups are elementary.
