In the symmetric group $S_3$, the task is to find elements α and β such that |α| = 2, |β| = 2, and |αβ| = 3. The elements of $S_3$ include three transpositions: (12), (13), (23) and two 3-cycles: (123) and (132), along with the identity element. By multiplying two transpositions, such as (12) and (23), the resulting product (12)(23) yields an element of order 3, satisfying the conditions of the problem.
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$S_3$ 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:
