  In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with
dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries
on $M$ with cohomogeneity at least $3$. After adapting the Almgren-Pitts
min-max theory to a $G$-equivariant version, we show the existence of a
nontrivial closed smooth embedded $G$-invariant minimal hypersurface
$\Sigma\subset M$ provided that the union of non-principal orbits forms a
smooth embedded submanifold of $M$ with dimension at most $n-2$. Moreover, we
also build upper bounds as well as lower bounds of $(G,p)$-width which are
analogs of the classical conclusions derived by Gromov and Guth. An application
of our results combined with the work of Marques-Neves shows the existence of
infinitely many $G$-invariant minimal hypersurfaces when ${\rm Ric}_M>0$ and
orbits satisfy the same assumption above.

Differential Geometry