What does the identity mean here?

[PhysicsRock]

Homework Statement

Let $S_3$ be the set of all bijections on the set $\{1,2,3\}$. Then, construct the Cayley table for the group $(S_3, id, \circ)$.

Relevant Equations

None.

I don't understand why the identity is mentioned in the group's definition and how I am supposed to incorporate it into the table. I honestly have missed some lectures on Linear Algebra, and I can't find any examples or definitions for this in the prof's notes. I'd appreciate some help for sure.

Thanks in advance.

 

[fresh_42]

It all depends on how you write those bijections. The notion $(S_3,id,\circ)$ is the most general one. It notes the set, the neutral element, and the binary operation.

Most common notation is to write those bijections as cycles: $S_3=\{(1),(12),(13),(23),(123),(132)\}$ where $(abc)$ stands for: $a\longmapsto b \longmapsto c \longmapsto a$ and $e\longmapsto e$ if a number isn't mentioned, or in case of $(1)$ where it is used instead of $().$ Here we have $(1)=id.$

 

[PhysicsRock]

It all depends on how you write those bijections. The notion $(S_3,id,\circ)$ is the most general one. It notes the set, the neutral element, and the binary operation.

Most common notation is to write those bijections as cycles: $S_3=\{(1),(12),(13),(23),(123),(132)\}$ where $(abc)$ stands for: $a\longmapsto b \longmapsto c \longmapsto a$ and $e\longmapsto e$ if a number isn't mentioned, or in case of $(1)$ where it is used instead of $().$ Here we have $(1)=id.$

Thank you so much. That helps a lot

 

[fresh_42]

You also must define whether you read consecutive bijections from left to right or from right to left. I prefer from right to left, i.e. $(f\circ g)[x]=f[g[x]].$ In the case of cycles, we get e.g. $(12)\circ (132)=(13).$ It reads $(12)\circ (132)[1]=(12)[3]=3\, , \,(12)\circ (132)[2]=(12)[1]=2\, , \,(12)\circ (132)[3]=(12)[2]=1$ so $1\longmapsto 3 \longmapsto 1$ and $2$ is a fixed point, i.e. the result is $(13).$