According to the Manual:

Any member can post reply to a forum post or start a forum topic in any public group, even if they aren’t a group member — adding a post or replying will auto join them to that group.

But then the Manual says later something that appears to indicate the opposite..
You can’t post a new Topic directly from the Forum Directory unless you are a member of a group(s).
When you add a new Topic on the Forum Directory you must select which Group you want the topic posted in.

I understand this is a requirement when the group is private.

My question is, can you create a new topic or forum within a public group, even if you are not a member of that public group, or one has to become a member of the group first?