I am using Fedora Linux

I created 2 users:
stewie : stewie, family
(stewie belongs to 2 groups)(stewie is the default group)
brian : users, family
(brian also belongs to 2 groups)(users is the default group)

Under root, I did the followings:
- Create directory /share/family
mkdir -p /share/family

- Change the group ownership recursively:
chgrp -R family /share/family

- Set gid bit on the /share/family dir:
chmod g+s /share/family

- Specify full permission for group family & no permissions for others
chmod -R g=rwx,o= /share/family

Then, at brian's terminal (after "su" to brian)
- I created a file: bri_f1. Check using "ls -l" -> it has only "r" permission for group family

Then, at stewie's terminal (after "su" to stewie)
- I created a file: ste_f1. Check using "ls -l" -> it has "rw" permissions for group family

So the 1st question is: why are the permissions different for files created by brian and stewie???

After that, at stewie's terminal, I modified the content of bri_f1 using "vi". I couldn't save the file because this is a read-only file. However, if I use :x!, I can override it. And if I "ls -l", the owner of the file bri_f1 has been changed to stewie (previously, it was of brian)

So, the 2nd question is: stewie doesn't have write permission on the file bri_f1; however, stewie can override it, and make the owner become himself. Why is that?

Thank you for any help.

Because brian and stewie (we aren't talking about Family Guy, right? ) are different users. I might missed something, so it would be better if you pasted exactly the output of the commands you ran in code tags.

stewie has write permissions on the directory containing the file, so he can delete/move/overwrite any file inside that directory. When you add "!" in your vim command, it doesn't write on the original file, it creates a new one discarding the brian's file.

Here is a recent thread that you should read on that topic.