Paper 2, Section I, E

Groups | Part IA, 2020

What does it mean for an element of the symmetric group SnS_{n} to be a transposition or a cycle?

Let n4n \geqslant 4. How many permutations σ\sigma of {1,2,,n}\{1,2, \ldots, n\} are there such that

(i) σ(1)=2?\sigma(1)=2 ?

(ii) σ(k)\sigma(k) is even for each even number kk ?

(iii) σ\sigma is a 4-cycle?

(iv) σ\sigma can be written as the product of two transpositions?

You should indicate in each case how you have derived your formula.

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