Facebook Groups cannot be used to promote a brand or product. They are meant to form organically out of your network of users using the social platform.
Another cosideration is the various setting of the groups such as open or closed group. If the group is closed, its content is not visible to the online public. Therefore, it definitely, will not show up in any search engines.
Then there is the much more engaging matter of search engine optimization. Google and Facebook have, to this date, not agreed on a clear system of how to rank Facebook’s content and this is due to the fact that facebook has so many privacy settings and other restrictions on the use of its content. So, for the most part, most facebook public content will place at the lowest end off the rankings, if at all.
Point to note here. Facebook has its own search capabilities using its edgerank algorithm. This, if used optimally, will provide higher ranking within facebook as it allows your content better visibilty to your members and in the case of Business Pages your fans.
Google is still the #1 search engine and as with any search engine, if you optimize the use of it, you will place higher in the search rankings and will be exposed to your potential customer or the person searching for your particular topic, much faster.