412.42 - Finding elements in S_3

[karush]

In $S_3$, find elements α and β such that |α| = 2,|β| = 2, and |αβ| = 3

 

[MarkFL]

I hate to be "this guy" again, but a thread title of "412.42" isn't very useful to the community. Please make sure thread titles briefly describe the question being asked. (Wave)

 

[Opalg]

In $S_3$, find elements α and β such that |α| = 2,|β| = 2, and |αβ| = 3

$S_3$ only has six elements, so you can list them all and do the question by trial and error. The elements consist of three transpositions ($(12)$, $(13)$ and $(23)$) and two 3-cycles ($(123)$ and $(132)$), the remaining element being the identity. Choose two elements with order 2, multiply them together and see whether the product has order 3.

 

[karush]

View attachment 8445

here is the example I think we are supposed to follow
but...

(123)(123)=?

 

[Opalg]

here is the example I think we are supposed to follow
but...

(123)(123)=?

The question is asking you to find two elements of order 2 whose product has order 3. So, what is the order of a 2-cycle and what is the order of a 3-cycle?

 

[karush]

so then
$$|\alpha\beta|=(12)(23)=3$$
?

 

[Opalg]

so then
$$|\alpha\beta|=(12)(23)=3$$
?

If you mean $|\alpha\beta|=|(12)(23)|=3$ then you're on the right track. But you'll need to specify the product $(12)(13)$, rather than just stating that it has order 3.

 

[karush]

ok, much mahalo,

this stuff is strange!