In mathematics: the braid group on strands, denoted by , is a group that generalizes the symmetric group . Here, is a natural number, representing a number of points to be permuted as strands. Each element of the symmetric group defines a permutation of these points from the initial to final configuration. An element of the braid group describes an initial and final configuration of these points, as well as how the stepwise configurations are composed by continuously moving the initial points to their final configurations. If , then is an infinite group.