I know that any finite group is isomorphic to a subgroup of the symmetric group of the same order. That got me thinking that finite braid groups (which I just recently learned are a thing) seem like they might be generated by a subset of all the "transpositions". Is that a concept even? Can I call the element of a braid group that just swaps the "landing" position of two strings a transposition?
2025-01-13 07:44:04.1736754244
Is any braid group generated by finite "transpositions"?
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Yes, these elements that swap two adjacent strands do generate the braid groups and finite braid groups. In the latter link, such elements are referred to as "elementary braids." Calling them "transpositions" may be confusing, because it's not clear that these correspond to transpositions in the isomorphic subgroup of $S_n$ (maybe they do, but it's not obvious to me).