On Oracle 9.2+ (not 8i+ as originally stated), SQL Server 2005+, PostgreSQL 8.4+, DB2, Firebird 3.0+, Teradata, Sybase, Vertica:
WITH summary AS (
SELECT p.id,
p.customer,
p.total,
ROW_NUMBER() OVER(PARTITION BY p.customer
ORDER BY p.total DESC) AS rk
FROM PURCHASES p)
SELECT s.*
FROM summary s
WHERE s.rk = 1
Supported by any database:
But you need to add logic to break ties:
SELECT MIN(x.id),
x.customer,
x.total
FROM PURCHASES x
JOIN (SELECT p.customer,
MAX(total) AS max_total
FROM PURCHASES p
GROUP BY p.customer) y ON y.customer = x.customer
AND y.max_total = x.total
GROUP BY x.customer, x.total