As title. This is question 52 in Chapter 5 of Gallian’s Abstract Algebra, 10th edition. My current line of logic is as follows.
We have $x^3 = (1234)$, which gives us $|x^3| = 4$, implying that $|x| = 12$. In $S_7$ there is only one orientation of $n-$cycles that has lengths with an lcm of 12, and that is $(a_1 a_2 a_3 a_4)(a_5 a_6 a_7)$. This works out, since we can have
$$(x)^3 = (\text{some rearrangement of 1, 2, 3, 4})^3(a_5 a_6 a_7)^3 = (\text{some rearrangement of 1, 2, 3, 4})^3 $$
My question, is how we get this correct rearrangement, and where do these three solutions come from, and what are they?
Thanks!
