Simplification of permutation $(1 2 3 4) (4 5 6 7)$

Question

Basic problem,

I'm confused on how to simplify this permutation into disjoint permutations?

$(1 2 3 4) (4 5 6 7)$

I am struggling between $(1 2 3 4 5 6 7)$ and $(1 2 3 5 6 7 4)$?

Thanks!

Answer 1

If you decompose these two permutations into product of transpositions, you will get: $$(1234)(4567)=(12)(23)(34)(45)(56)(67),$$ so from here you can notice that this is the decomposition into transpositions of the permutation $(1234567)$, so we conclude that $$(1234)(4567)=(1234567).$$

If you want to calculate it directly, you have to take a look to where does each number go (doing the permutations from right to left, so the first arrow points to the number after the right permutation and the second points to the number after the left permutation): $$ 1 \to 1\to 2$$ $$ 2 \to 2\to 3$$ $$ 3 \to 3\to 4$$ $$ 4 \to 5\to 5$$ $$ 5 \to 6\to 6$$ $$ 6 \to 7\to 7$$ $$ 7 \to 4\to 1$$ And, as you see, this gives us the same permutation result: $(1234567)$.