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MellyVG257
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Homework Statement
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.
Homework Equations
The Attempt at a Solution
What I was actually looking for is where to start with this proof. I don't want the answer, just a push in what direction I should be heading in. This is my trouble with proofs, I usually have no idea where to start.
I've been told in the past to start with definitions of what is given to you in the question.
A transposition is a permutation (bijective function of X onto itself) f, such that there exist i,j such that f(a_i) = a_j, f(a_j) = a_i and f(a_k) = a_k for all other k.
I know that "l" is the length of the cycle.
I also 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? Sorry, I'm just having a hard time with understanding this one.
But I don't see how this helps me. Any suggestions?
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