Question

How to find order of permutation (4568)(1245) in s8?

Answer 1

=(14223941056657118798101113)
σ=(12345678910114291065117813)
(1) I am asked to write this permutation in S11S11 as a product of disjoint cycles and also as a product of transpositions.

(2) Also find the order of the element. Is this permutation even or odd?

i think these are the disjoint cycles

E1=(1,4,10)E1=(1,4,10), orderE1=3order⁡E1=3
E2=(3,9,8,7,11),E2=(3,9,8,7,11), orderE2=5order⁡E2=5
E3=(5,6),E3=(5,6), orderE3=2order⁡E3=2
S11S11= E1⋅E2⋅E3E1⋅E2⋅E3
orderE3order⁡E3 is even so the order of the permutation is even. Why are they asking this? and what is the significance of it being even or odd?

transpositions i have read this a couple times but the one example in my textbook is rather unclear, i am not sure what this means?

i think the transposition for E1E1 is (1,4)(1,10)(1,4)(1,10) but im not sure what this means.