The discussion revolves around finding elements α and β in the symmetric group $S_3$ such that |α| = 2, |β| = 2, and |αβ| = 3. The scope includes mathematical reasoning and exploration of group theory concepts.
Participants generally agree on the approach of using trial and error with the elements of $S_3$, but there is some uncertainty regarding the specific products and their orders.
Some participants express confusion about the properties of the elements and the calculations involved, indicating a need for clearer definitions and understanding of group elements.
Readers interested in group theory, particularly those studying symmetric groups and their properties, may find this discussion relevant.
$S_3$ 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 said:
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 said:
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 said:
