- #1

- 2

- 0

I have no idea what im supposed to do with it, i know why S3 has only one element that commutes but i dont know how to prove it for all.

- Thread starter emath
- Start date

- #1

- 2

- 0

I have no idea what im supposed to do with it, i know why S3 has only one element that commutes but i dont know how to prove it for all.

- #2

- 22,089

- 3,293

If f and g are in S

Think how that would prove your claim.

- #3

jgens

Gold Member

- 1,581

- 50

If [itex]\sigma \in S_n \setminus \{\mathrm{id}\}[/itex], then there exists an [itex]i \in \{1,\dots,n\}[/itex] such that [itex]\sigma(i) = j[/itex] and [itex]i \neq j[/itex]. Now let [itex]k \in \{1,\dots,n\}[/itex] be such that [itex]k \neq j,\sigma(j)[/itex] and let [itex]\tau \in S_n[/itex] be the transposition which switches [itex]j[/itex] and [itex]k[/itex]. The rest of the proof is trivial from here.

- Replies
- 6

- Views
- 940

- Replies
- 2

- Views
- 488

- Replies
- 4

- Views
- 637

- Replies
- 3

- Views
- 4K

- Replies
- 1

- Views
- 2K

- Last Post

- Replies
- 5

- Views
- 708

- Replies
- 7

- Views
- 3K

- Last Post

- Replies
- 3

- Views
- 4K

- Replies
- 10

- Views
- 9K

- Replies
- 3

- Views
- 3K