  We introduce the classical theory of the interplay between group theory and
topology into the context of operads and explore some applications to homotopy
theory. We first propose a notion of a group operad and then develop a theory
of group operads, extending the classical theories of groups, spaces with
actions of groups, covering spaces and classifying spaces of groups. In
particular, the fundamental groups of a topological operad is naturally a group
operad and its higher homotopy groups are naturally operads with actions of its
fundamental groups operad, and a topological $K(\pi,1)$ operad is characterized
by and can be reconstructed from its fundamental groups operad. Two most
important examples of group operads are the symmetric groups operad and the
braid groups operad which provide group models for $\Omega^{\infty}
\Sigma^{\infty} X$ (due to Barratt and Eccles) and $\Omega^2 \Sigma^2 X$ (due
to Fiedorowicz) respectively. We combine the two models together to produce a
free group model for the canonical stabilization $\Omega^2 \Sigma^2 X
\hookrightarrow \Omega^{\infty} \Sigma^{\infty} X$, in particular a free group
model for its homotopy fibre.
