# Monodromy group

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The **monodromy group** of an abstract polytope, 𝓟, written Mon(𝓟), the group generated by flag changes.

## Definition[edit | edit source]

For a polytope 𝓟 of rank n , let σ i be the bijection which maps each flag of 𝓟 to its i -adjacent flag. Then the monodromy group is ⟨σ 0 ,σ 1 ,...,σ n -1 ⟩, that is the group generated by flag changes.

## Properties[edit | edit source]

- A polytope is regular iff its monodromy group is isomorphic to its automorphism group. Furthermore there is an isomorphism between the two which maps the flag changes to the distinguished generators.
- The monodromy group of a polyhedron is the automorphism group of its minimal regular cover.
^{[1]}In ranks greater than 3 this is not always the case - the tomotope is an example.

## References[edit | edit source]

## Bibliography[edit | edit source]

- Monson, Barry; Pellicer, Daniel; Williams, Gordan (2012), "The Tomotope",
*Ars Mathematica Contemporanea*

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