IDA: Interactive Document on Algebra

Cycles are elements in S_n of special importance.

Definition

Let g S_n be a permutation with supp(g) = {a₁, ..., a_m}, where the a_i are pairwise distinct. We say g is an m-cycle if g(a_i) = a_i+1 for all 1 i < m and g(a_m) = a₁. For such a cycle g we also use the cycle notation

g = (a₁, a₂, ..., a_m).

2-cycles are called transpositions.

The m-cycle g = (a₁, a₂, ..., a_m) shifts the a_k cyclically:

It is evident that the square of this permutation maps a₁ to a₃, a₂ to a₄, etc. (What happens if m = 2 or 3?) In a similar way you can figure out what the other powers of g are. The order of an m-cycle is m.