The order of a permutation w is the minimal positive integer k so that w^k is the identity. (This is the same as the order of w in the multiplicative group of permutations). Find the order of the following permutations in S_n:
1. (1234) when n=4
2. (12345) when n=5
3. (123...r)(r+1)(r+2)...(n) when r is at most n
4. (1234)(567) when n=7
5. (12345)(67) when n=7
6. (1234)(56) when n=6
Any conjectures? Can you prove any of your conjectures?
An element g in a (multiplicative) group G is generated by the elements g₁, g₂, ..., g_k if g can be written as a product of the g_i and their inverses. For instance (13) is generated by (12) and (23) because (13)=(12)(23)(12). The group G is generated by the elements g₁, g₂, ..., g_k if every element in G is generated by g₁, g₂, ..., g_k.
1. Find a set of generators for S₃. (Prove.)
2. Find a set of generators for S₄. (Prove.)
3. Can you find a set of generators for S_n? Can you prove it?
4. Can you find a different set of generators fof Sn? Can you prove it?