The Student Room Group

Permutation Group S3

Ok just wondering what the permutation group for S3 is. My real question is do the elements move the entries or the position of the entires?

i.e.

I have the results of the permutations:

{(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)} \{ (1,2,3) , (1,3,2) , (2,1,3) , (2,3,1) , (3,1,2) , (3,2,1) \}

and now I want to assign the elements of S3 to these as follows:

S3={(e),(23),(12),(A),(B),(13)} S_3 = \{ (e) , (23) , (12) , (A) , (B) , (13) \}

what are A and B??

If the elements move the positions then I have:

(A)=(12)(23),(B)=(12)(13) (A) = (12)(23) , (B) = (12)(13)

however if they move the entries then we have the opposite:

(A)=(12)(13),(B)=(12)(23) (A) = (12)(13) , (B) = (12)(23)

either way I know what the group is, but I don't know what the elements are doing. Are they acting on the position or the entry??

An easy way to look at it might be, if we have the results of S3 as:

{(a,b,c),(a,c,b),(b,a,c),(b,c,a),(c,a,b),(c,b,a)} \{ (a,b,c) , (a,c,b) , (b,a,c) , (b,c,a) , (c,a,b) , (c,b,a) \}

are the elements of S3 written in terms of a, b, c or are they written in terms of 1,2,3??

Thanks.
Reply 1
Avatar for sputum
sputum
9
In your permutation 'set' you have (a,b,c) and (b,c,a). These are the same permutation. As is (c,a,b).

(a,c,b) = (b,a,c) = (c,b,a)

so you only have two different permutations.

The elements of S3 are written so that (12) = (12)(3) (a permutation that sends 1 to 2, 2 to 1 and 3 to 3) for example.
Reply 2
Avatar for adie_raz
adie_raz
OP
1
Original post by sputum
In your permutation 'set' you have (a,b,c) and (b,c,a). These are the same permutation.


What I wrote above is not supposed to be the permutation set. It is the results from the permutation set. I want to write the permutation set corresponding to these results.

Say I start with (a,b,c) and I end with (b,c,a) then the permutation in the permutation set is what??

(12)(23) or (ab)(ac) or neither??

Thanks for your reply.
Reply 3
Avatar for sputum
sputum
9
Oh the penny has dropped, sorry:smile:
So your first (1,2,3) corresponds to the function
1 -> 1
2 -> 2
3 -> 3
the second to
1 -> 2
2 -> 1
3 -> 3
etc (I hope)

(a,b,c) and I end with (b,c,a)
is
a->b
b->c
c->a

Standard permutation notation would just record this as (123) and it is a good candidate for one of your missing entries.
Reply 4
Avatar for adie_raz
adie_raz
OP
1
Original post by sputum
Oh the penny has dropped, sorry:smile:
So your first (1,2,3) corresponds to the function
1 -> 1
2 -> 2
3 -> 3
the second to
1 -> 2
2 -> 1
3 -> 3
etc (I hope)

(a,b,c) and I end with (b,c,a)
is
a->b
b->c
c->a

Standard permutation notation would just record this as (123) and it is a good candidate for one of your missing entries.


Got it thanks, what I was missing was:

(A) = (123) , (B) = (132).

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you