# What is the name of the symmetry of a bracelet transposition?

Take a bracelet with colored beads on it. Normally two bracelets belong to the same equivalence class under rotations and reflections. For an example, consider the bracelet denoted by the word abcdec:

cabcde ecabcd decabc cdecab ... (rotations)
cedcba dcbace ... (reflections)


Consider the larger class where two bracelets are equivalent if all local bead pairings are preserved. From our example we have the multiset {ab, bc, cd, de, ec, ca}. The right-left ordering, ba vs ab doesn't matter. Clearly rotations and reflections preserve bead pairings, but so does:

abcedc


where we've transposed the e and the d. How do you describe this symmetry?

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If it helps, your bead pairings are the edges of a multi-graph, and a word in the equivalence class is the same thing as a Euler circuit of the graph. –  Hurkyl Jun 10 '12 at 5:42