Clearer Understanding of Permutation and Transpositions

[MellyVG257]

Let α (alpha) all in S_n be a cycle of length l. Prove that if α = τ_1 · · · τ_s, where τ_i are transpositions, then s geq l − 1.

I'm trying to get a better understanding of how to begin proofs. I'm always a little lost when trying to solve them.
I know that I want to somehow show that s is greater than l - 1 cycles. Does this mean I need to find out or show that any l cycle can be written as a product of l-1 cycles? I wrote that

α = τ_1 · · · τ_s = (τ_1 τ_s)(τ_1 τ_s-1)...(τ_1 τ_2)

But does this qualify as a proof for showing that any l cycle can be written as a product of l-1 cycles? Even so, how does this make sense for s geq l -1? Sorry, I'm just trying to understand this more clearly.

 

[Jösus]

I'm not sure I understand your formulation. Or, it is your partial solution I do not grasp. Do I interpret i correctly if I'd restate it as follows?

Let a be a k-cycle in Sn. Show that if a = t1 * t2 * ... * ts, where the ti are transpositions for 1 = 1, 2, ..., s, then s must be greater than or equal to k - 1.

We have that, if a = (A1, A2, ... ,Ak), then a = (A1, Ak)(A1, Ak-1)...(A1, A2).

Well, this is not a proof, as it only states that there exists a way to express a as a product of k - 1 transpositions. Now, it is easily seen that we can express a as a product of a larger number of such transpositions. Just multiply it by (1, 2)(1, 2), which is the identity. The problem is to show that there is no way to express a as a product of less transpositions. I haven't proven it, but I got a feeling that thinking about how a "moves" the Ai can lead you in the right direction.