A finitely generated group is a (discrete) group (equipped) with a finite set of generators, hence of elements such that any other element can be written as a product of these.
Associated with a set of generators on a group is the corresponding Cayley graph, whose vertices are the group elements which are connected by an edge if they differ by (left, say) multiplication with one of the generators.
The corresponding graph distance equips the group with a metric (the word metric) and thus makes it an object of geometric group theory.
(symmetric group)
The symmetric group $Sym(n)$ of permutations of $n \in \mathbb{N}$ elements may be generated from:
all transposition permutations – the corresponding Cayley graph distance is the original Cayley distance;
the adjacent transpositions – the corresponding Cayley graph distance is known as the Kendall tau distance.
(braid group)
The braid group $Br(n)$ on $n \in \mathrm{N}$ strands is finitely generated via the Artin presentation.
Textbooks:
See also
Discussion in homotopy type theory/univalent foundations of mathematics:
Last revised on December 23, 2022 at 23:29:43. See the history of this page for a list of all contributions to it.