- #1
PsychonautQQ
- 784
- 10
Homework Statement
I'm taking an online class this summer and the notes gave the proposition that Disjoint Cycles Commute with the following proof.
Proof. Let σ and τ represent two disjoint cycles in Sn and choose some arbitrary
j ∈ {1,2, . . . , n}. Since σ and τ are disjoint, at most one of them fixes j, so suppose τ
fixes j. But then τ must also fix σ.j since the cycles are disjoint. Hence στ.j = σ.k and
τσ.j = σ.j.
What does this proof mean when it says that it "fixes" j? what exactly does commute mean again? can anyone help me make sense of this?