Joining a research group is one of the most important events on a graduate student's path to earning a PhD, but the ways students go about searching for a group remain largely unstudied. It is therefore crucial to investigate whether departments are equitably supporting students as they look for an advisor, especially as students today enter graduate school with more diverse backgrounds than ever before.  To better understand the phenomenon of the group search process, we use a comparative case study approach to contrast important aspects of two physics PhD students' experiences.  Semi-structured interviews with the students chronicled their interactions with departments, faculty, and the graduate student community, and described the resources they found most and least helpful.  Our results reveal significant disparities in students' perceptions of how the group search process works, as well as inequities in resources that negatively influenced one student's experience.  We also uncover substantial variation regarding when in their academic careers the students began searching for a graduate advisor, indicating the importance of providing students with consistent advising throughout their undergraduate and graduate experiences.