there are no "read only" groups
there are files/folders that have read only permission for a group
if a group has no write access to any file on a system, you may call it read only group, but there's still the /tmp folder, where everyone has write rights
normally a new created user just has write rights to his home directory
if you change the home directory of the user to something like
chmod 555 /home/user , the user should be a read only user for the most directories on your system
but any time you change a directory to 777 for example, the user will be able to write to that directory, so be careful
perhaps you want to have a look at:
UNIX permissions