SQL Query: Selecting a person who is a member of all existing groups
There are several ways to get to this answer.  I'll tackle it in two parts.  The first is to find out how many groups there are.  This is a simple summary query:

select count(distinct GroupID)
from MEMBER


The second is how many groups one person is in.  This is similar:

select
  PID,
  COUNT(*)
from MEMBER
group by PID


Next you want to tie these together

select
  PID,
  COUNT(*)
from MEMBER
group by PID
having COUNT(*) = (select count(distinct GroupID)
                   from MEMBER)


Depending on which RDBMS you're using (Oracle, DB2, MySQL) there may be other syntaxes or layouts which make this faster, prettier or otherwise more desirable.