ids from grouped rows

Lets say i do this:

SELECT name, MAX(age)
FROM Person
GROUP BY name

This will group people who have the same name together, and return the
highest age for each group. I want to be able to find the id for the
person with the highest age in each group  -

Ideally, it would be something like this

SELECT name, MAX(age), id_for_row_with_max_age
FROM Person
GROUP BY name

Anyone know how to do this?

Lindsay