>Let $ \sigma_{2} = (4215)(3426)(5617)$. Show that permutation as products of, in pair, disjoint cycles and as product of transpositions.

Question

Let $ \sigma_{2} = (4215)(3426)(5617)$. Show that permutation as products of, in pair, disjoint cycles and as product of transpositions.

I am a beginner and I am not sure how to start. Any hint helps! Thanks.

Answer 1

First the decomposition as a product of disjoint cycles: compute the final image of $1$, then the image of its image, and so on:

• $1\mapsto7$,
• $7\mapsto 5\mapsto 4$,
• $4\mapsto2\mapsto 1$.

So we have a first cycle: $\;(1\,7\,4)$.

Can you calculate now the cycle starting with $2\,$?

As to expressing a cycle as a product of transpositions, it is easy: consider, for instance, the cycle $(1\,4\,3\,2)$. Multiply it on the left by the transposition $(1\,4)$: $$(1\,4)(1\,4\,3\,2)=(4\,3\,2).$$ Multiply the result by $(4\,3)$: $$(4\,3)(4\,3\,2)=(3\,2)$$ Thus we obtain $$(4\,3)(1\,4)(1\,4\,3\,2)=(3\,2),$$ and using that a transposition is its own inverse: $$(1\,4\,3\,2)=(1\,4)(4\,3)(3\,2),$$ from which you can infer an easy computation rule for the decomposition of a cycle as a product of transpositions.

Answer 2

Start with $1$ and see where it goes, by applying the permutations from right to left. Get $1\to7\to4\to1$, so $(174)$. Then $2\to6\to5\to3\to2$, so $(2653)$.

Next one way of writing a cycle as a product of transpositions is, for instance: $(1234567)=(17)(16)(15)(14)(13)(12)$.