Why does $(13)(47)(13)(23)=(47)(23)$?

Question

Given $a=(3714)$ and $b=(123)$. Compute $a^2b$. I got $(13)(47)(13)(23)$ as my answer, but why does $(13)(47)(13)(23)=(47)(23)$? I got the left hand side of this equation, but I don't understand why the $(13)$ terms cancel out. Any help is greatly appreciated.

Answer 1

Because disjoint cycles commute, as others have already said. But to convince yourself of this, it might help to follow what happens to each of the elements $1,2,3,4,7$ as they "go through the mill". For instance, look what happens to $3$: $$(13): 3\to 1$$ $$(47): 1\to 1$$ $$(13): 1\to 3$$ $$(23): 3\to 2$$ So $3$ gets mapped to $2$.

If you do this for yourself, you will find that the result is the same as $(47)(23)$.

Answer 2

Becouse this commutes: $(13)(47)(13)$; also if you check exactly where the elements go, you get that it is the same as the right side.

Answer 3

The key insight that you need is that disjoint cycles commute. This means that:

$$ (1 3)(4 7)(1 3)(2 3) = (1 3)(1 3)(4 7)(2 3) = (4 7)(2 3) $$

Answer 4

Hint: $(13)$ and $(47)$ commute, because they are disjoint. Also, $(13)$ has order two.