Despite the prominence of analogies in mathematics, little attention has been given to exploring students' processes of analogical reasoning, and even less research exists on revealing how students might be empowered to independently and productively reason by analogy to establish new (to them) mathematics. I argue that the lack of a cohesive framework for interpreting students' approaches to analogical reasoning in mathematics contributes to this issue. To address this, I introduce the Analogical Reasoning in Mathematics (ARM) framework. Constructed from an analysis of interviews with four abstract algebra students, ARM identifies several analogical activities that serve to analyze students' analogical reasoning with a finer grain size than was previously possible with existing frameworks. Using this framework, I present an analysis of the students' constructions of a ring-theoretic analogy to subgroup, thus revealing that even constructing simple analogies can elicit diverse pathways of analogical reasoning across students. Implications for further research related to analogies and analogical reasoning in mathematics education are discussed.