G^N vs Lie group

G^N is a special calculator not just a game: This math app animates iterations of one or compositions of two multi-valued maps for free groups of in principle arbitrary finite rank in the abelian and non-commutative cases respectively (in this version the rank is restricted to values in between 2 and 11). Well-known examples of such multi-valued maps are permutations, for example the flip (x_1,x_2)->(x_2,x_1), in G^N we just write (1,2)->(2,1) and instead of the inverse we convenient flip the number. A more involved type of such a multi-valued map is the n+1 cycle defined by (1,...,n)->(-2-3-...-n-1,3,4,...,n,1), a non-trivial theorem. For example if n=2 this map is the hexagon (1,2)->(-2-1,1), this map applied twice reads (1,2)->(2,-2-1) and the third iteration is the identity map (1,2)->(1,2).

This app represents Lie algebra and Lie group. You can have some understanding for Lie algebra and Lie group.