Putting the "rw" in the "group area" makes it readable and writable by the assigned group. The part your missing is assinging the group to the file.
Do ls -l <filename>
This will show you the permissions for user, group and other, the number of links, the owner (user for permissions) of the file and the group of the file as well as other information. So looking at permissions:
The first position would indicate the file type (- = regular file, d = directory, l = symbolic link, c = character device, b = block device etc...). The next 3 describe the permissions for the owner seen on the line. The next 3 describe the permissions for the group on the line and the last 3 describe the permissions for everyone else.
r=read (value = 4), w=write (value=2), x=execute (value=1).
So rwx = 7, rw- = 6, r-x = 5 etc...
You change the group of a file with the chgrp command. You change the owner with the chown command. (You can set both with chown by typing chown <owner>:<group> <filename> where owner is the owner you want, group is the group you want and filename is the file you want to change.)