The quotient group concept is a difficult for many students getting started in abstract algebra (Dubinsky et al., 1994; Melhuish, Lew, Hicks, and Kandasamy, 2020). The first study in this thesis explores an undergraduate, a first-year graduate, and second-year graduate students' representational fluency as they work on a "collapsing structure", quotient, task across multiple registers: Cayley tables, group presentations, Cayley digraphs to Schreier coset digraphs, and formal-symbolic mappings. The second study characterizes the (partial) make-up of two graduate learners' example-based intuitions related to orbit-stabilizer relationships induced by group actions. The (partial) make-up of a learner's intuition as a quantifiable object was defined in this thesis as a point viewed in R17, 12 variable values collected with a new prototype instrument, The Non-Creative versus Creative Forms of Intuition Survey (NCCFIS), 2 values for confidence in truth value, and 3 additional variables: error to non-error type, unique versus common, and network thinking. The revised Fuzzy C-Means Clustering Algorithm (FCM) by Bezdek et al. (1981) was used to classify the (partial) make-up of learners' reported intuitions into fuzzy sets based on attribute similarity.