Abstract

Wilhelm (Forthcom Synth 199:6357–6369, 2021) has recently defended a criterion for comparing structure of mathematical objects, which he calls Subgroup. He argues that Subgroup is better than SYM $^*$, another widely adopted criterion. We argue that this is mistaken; Subgroup is strictly worse than SYM $^*$. We then formulate a new criterion that improves on both SYM $^*$ and Subgroup, answering Wilhelm’s criticisms of SYM $^*$ along the way. We conclude by arguing that no criterion that looks only to the automorphisms of mathematical objects to compare their structure can be fully satisfactory.