Lemma 58.4.2. In diagram (58.4.0.1) the following are equivalent

$h \circ h'$ is trivial, and

the image of $H' \circ H$ consists of objects isomorphic to finite coproducts of final objects.

Lemma 58.4.2. In diagram (58.4.0.1) the following are equivalent

$h \circ h'$ is trivial, and

the image of $H' \circ H$ consists of objects isomorphic to finite coproducts of final objects.

**Proof.**
We may replace $H$ and $H'$ by the canonical functors $\textit{Finite-}G\textit{-Sets} \to \textit{Finite-}G'\textit{-Sets} \to \textit{Finite-}G''\textit{-Sets}$ determined by $h$ and $h'$. Then we are saying that the action of $G''$ on every $G$-set is trivial if and only if the homomorphism $G'' \to G$ is trivial. This is clear.
$\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)