In 2013, Montaldi and Steckles categorized all possible symmetry groups of planar 𝑛-body collision-free choreographies, which can be divided into two infinite families and, in the case of odd values of 𝑛, three exceptional groups. The second part of their study involves the development of the equivariant fundamental group, which is employed to identify the topology of the space of loops possessing a particular symmetry. They demonstrate that this topology is linked to certain cosets of the pure braid group in the full braid group, as well as the centralizers of elements within the corresponding coset. Furthermore, their work refines the symmetry classification by categorizing the connected components of the set of loops with a given symmetry, leading to the discovery of many new choreographies in 𝑛-body systems that are governed by a strong force potential.