When a member of multiple groups, how to set the default group for newly created files

Folder b and c are owned by user b and c. A file created by a user will belong to that user.

You can use the user permission for b and c, and the group permissions for a. If you set the SGID bit (g+s) on a folder, created files will get the group permission of that folder.

mkdir a
chown a:a a
chmod g+s a

mkdir b
chown b:a b

mkdir c
chown c:a c

(assuming all users are in a group of the same name.)

Yes, you can do that changing the umask. The umask determines which are the default permissions for a newly creted file.

You can add umask g+w at the end of your shell configuration file (~/.bashrc for example).

But actually, it´s not a recommendable practice. In the case you do want to ensure the integrity of a file and you forget to update the file permissions, it will be modifiable by the group. It's against the "secure initial values" principle of security.

What you could do instead is make all the newly created file of a specific directory writable by the group. You can do this manipulating the ACLs of the directory. For example, setfacl -dm u::rw,g::rw,o::r ~/shared.

Look at those posts for reference : https://serverfault.com/questions/349145/can-i-override-my-umask-using-acls-to-make-all-files-created-in-a-given-director and https://stackoverflow.com/questions/580584/setting-default-permissions-for-newly-created-files-and-sub-directories-under-a.