I am using a Mac with OS X and have installed mySQL. However I have some type of security issue which blocks me. I have read the various forums and advice given there fails.

Two extracts of my issue:

1./
sql> create database A;
error 1044 : access denied for user : '@localhost' to database 'A'

2./
the unix command ls -l on the mysql directory gives "drwx-------- root data" for the data directory.

I think this is the issue. Maybe not.

I am no unix specialist and innocently thought that mac would fend-off me having to return to my university books to start a database system!

Can someone help?

the problem is that you are logged with an anonymous user that got no privilege to create a database.
You should use root or other user with create db privilege