Question

Is there a difference between the permutation (1 2 3 4)(4 5 6) in S_7 as oppose to if it were in S_8?

Answer 1

@phi @SithsAndGiggles @jim_thompson5910 @satellite73

Answer 2

In S_7, that permutation leaves 7 fixed. In S_8, it leave both 7 and 8 fixed. Thats the other difference.

Answer 3

only* difference

Answer 4

The standard notation for a permutation leaves out any numbers that aren't permuted. If we put those back in the differences become clearer. So in S_7, your permutation is: $$(1234)(456)(7),$$and in S_8: $$(1234)(456)(7)(8)$$

Answer 5

Thank you! That's what I thought, but I wasn't sure since I figured the 8 had to be the order of the cycle