Many results in group theory were proven for permutation groups before the general definition of a group was established (for example: Lagrange's theorem, Sylow's theorems). However, permutation groups weren't the only groups being studied in the nineteenth century, there were also groups of geometric transformations and group arising from number theory (I can't really give more details because I don't frankly know the details).
Were any general group theory results known for several specific cases other than just permutation groups, before the general definition of a group had been formulated? I ask because I'm wondering if such "coincidences" could have motivated the general definition of a group. As an example, Lagrange's theorem was known in the 19th century both for permutation groups and for the multiplicative group of $\mathbb Z/n\mathbb Z$ (via Euler).