by [@Go-Ma Prop. 2.6], any transitive action is Cohen-Macaulay, without being in general equivariantly formal). However, the following result is helpful in this context, see [@Go-Ro Prop. 2.5]:

\[characef\] A smooth action $G\times M \to M$ is equivariantly formal if and only if it is Cohen-Macaulay and there exists at least one point in $M$ whose isotropy subgroup has rank equal to ${\rm rank } \ \! G$.

For later use, we also mention:

\[maxtor\] Let $G\times M \to M$ be a smooth action and $T\subset G$ an arbitrary maximal torus.

\(a) The $G$-action on $M$ is equivariantly formal if and only if so is the induced $T$-action.

\(b) The $G$-action on $M$ is Cohen-Macaulay if and only if so is the induced $T$-action.

Item (a) is a standard result, see for instance [@GGK Prop. C.26]. Item (b) is the content of [@Go-Ro Prop. 2.9]. It is worth pointing out in this context that if $H\subset G$ is an arbitrary closed subgroup, then the property of being equivariantly formal is preserved when passing from $G$ to $H$: indeed, on the one hand, the $G$-action is equivariantly formal if and only if the canonical map $H^*_G(M)=H^*(E\times_G M) \to H^*(M)$ is surjective 