Symmetric group S3 with symbols

[mikael27]

Homework Statement

Determine the orders of all the elements for the symmetric group on 3 symbols S3.

Homework Equations

_______________________________________

The Attempt at a Solution

3 symbols : e,a,b

I don't know how to do the S3 table using just these 3 letters
I can do it using 6 letters (e,s,t,u,v,w) which is going to be like this:

e s t u v w
s e v w t u
t u e s w v
u t w v e s
v w s e u t
w v u t s e

 

[mikael27]

Homework Statement

Determine the orders of all the elements for the symmetric group on 3 symbols S3.

Homework Equations

3. The Attempt at a Solution [/b

3 symbols : e,a,b

I don't know how to do the S3 table using just these 3 letters
I can do it using 6 letters (e,s,t,u,v,w) which is going to be like this:

e s t u v w
s e v w t u
t u e s w v
u t w v e s
v w s e u t
w v u t s e

 

[Deveno]

the "symbols" of S₃ in this case aren't the elements, they are the elements of the domain of the functions that make up S₃.

ok, what S₃ is, is the group of permutations of 3 "things". sometimes the things are called "letters", sometimes they are taken to be the elements of the set {1,2,3}. but what an element of S3 IS, is a certain kind of function: namely, a bijection on a 3-element set with itsef. all 3-element sets are pretty much alike (as sets), they just have different "element-names". so it doesn't really matter if you call the element (1 2) in S₃ the function:

1→2
2→1
3→3

or:

a→b
b→a
c→c

what we actually mean by (1 2) is:

"the function on a 3-element set that swaps the first and second elements".

there are 6 possible bijections on the 3-element set A = {a,b,c} (you can think of a = 1, b= 2, and c = 3 if you want to, but the properties of 1,2 and 3 as integers aren't relevant here. it's better to think of them as:

a = "element number 1"
b = "element number 2"
c = "element number 3").

one of the 6 possible bijections is:

a→a
b→b
c→c

better known as: the identity function on A: f(x) = x, for all x in A.

there are 3 bijections (permutations) of A that interchange 2 elements, and 2 bijections of A that send all 3 "to something else". see if you can puzzle out what these mappings might be. then give all 6 maps a name.

multiplication on S₃ is functional compostion. so if f is in S₃, you want to answer the following question:

"what is the smallest positive integer n, for which f°f°...°f(x) = fⁿ(x) = x, for all x in A?"

that integer is the order of f.

for our example f =

a→b
b→a
c→c,

f² =

a→b→a
b→a→b
c→c→c

so f²(x) = x, for all x in A, that is: f² = id_A, so the order of f is 2, in this case. 1 down, 5 to go.

 

[HallsofIvy]

I have merged the two threads. Please do not post the same question more than once.