Understanding Disjoint Cycles and Commutativity in Sn

[PsychonautQQ]

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?

Homework Equations

The Attempt at a Solution

 

[pasmith]

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?

$\sigma$ fixes $j$ if and only if $\sigma(j) =j$.

what exactly does commute mean again?

$\sigma$ and $\tau$ commute if and only if for all $j \in \{1, 2, \dots, n\}$, $\sigma(\tau(j)) = \tau(\sigma(j))$.

 

[bloby]

Since σ and τ are disjoint, at most one of them fixes j, so suppose τ
fixes τ.

At most or at least?

 

[PsychonautQQ]

The notes say at most, but there have been errors before, i'll bring it up with him at our next meeting. Thanks guys :D