Doubts from transposition, fixing and Identity

[PcumP_Ravenclaw]

Dear All,
Please see the attachment for the text i will be referring to.

what does "I = τ1 · · · τm, where each τj is a transposition acting on {1, . . . , n}. Clearly, m != 1, thus m ≥ 2. Suppose, for the moment, that τm does not fix n".

what does fixing n mean in a transposition?

what are a.b and c? I thought all transpositions started with 1?

what does this whole sentence mean "It follows that we can now write I as a product of m transpositions in which the first transposition to be applied fixes n (this was proved under the assumption that τm(n) != n, and I is already in this form if τm(n) = n)." ?

please help me understand lemma 1.4.3 with an example??

Thanks...

 

[Stephen Tashi]

Let's start with this:

what are a.b and c? I thought all transpositions started with 1?

Why did you think that? How does the text define a transposition? Are you reading the definitions in the text carefully or are you making up definitions for things in you own words? Mathematics is legalistic. The definitions mean exactly what they say. You have to approach them in a nit-picking manner. If you scan a definition and substitute your own words for the meaning, you will go far astray.

please help me understand lemma 1.4.3 with an example??

Try to start the example by yourself. Begin by writing the identity function in some way as a product of transpositions. Pick an example more elaborate than writing the identity as (2,3)(3,2).