Is it possibly considered weakness in my statement of purpose if I mention that I would like specific people to advise my PhD programme? I'm writing a statement of purpose for my PhD applications and in some cases I'd be very happy if any person from a certain group (say the Algebra Group) would take me as his/her student. Could this be considered a weakness in my application? How should I handle this?

I've been researching people and groups for over 4 months and I know (as much as I can right now) what I want to do and who does it. It's just that some universities and programs have some amazing faculty.
I would include which certain aspects of the faculty would support your intentions and why, especially if you can include meaningful references to their previously published work.

For example

"I believe that University X would be a great place to undertake my research into topic Y because it has a great reputation in this field. For example books A and B published by person C are well known texts in the field and I believe the research of person D in the area of E would compliment mine."