Write $(12)(13)(23)(142)$ as the product of disjoint cycles.

Question

I got that

$$(12)(13)(23)(142) = (14)(2314) = (234)$$

I am thinking that this is incorrect, but I am unsure how to fix my work. Can someone tell me if this is the correct approach?

Thanks!

Answer 1

We have

$$\begin{align} 1&\stackrel{(142)}{\mapsto}4\stackrel{(23)}{\mapsto}4\stackrel{(13)}{\mapsto}4\stackrel{(12)}{\mapsto}4, \\ 2&\stackrel{(142)}{\mapsto}1\stackrel{(23)}{\mapsto}1\stackrel{(13)}{\mapsto}3\stackrel{(12)}{\mapsto}3, \\ 3&\stackrel{(142)}{\mapsto}3\stackrel{(23)}{\mapsto}2\stackrel{(13)}{\mapsto}2\stackrel{(12)}{\mapsto}1, \\ 4&\stackrel{(142)}{\mapsto}2\stackrel{(23)}{\mapsto}3\stackrel{(13)}{\mapsto}1\stackrel{(12)}{\mapsto}2, \end{align}$$ so the product is $(1423)$.

Answer 2

$(12)(13)(23)(142)=(1423)$, just by checking where each element goes.