Paper 3, Section II, E

Groups | Part IA, 2017

Prove that every element of the symmetric group SnS_{n} is a product of transpositions. [You may assume without proof that every permutation is the product of disjoint cycles.]

(a) Define the sign of a permutation in SnS_{n}, and prove that it is well defined. Define the alternating group AnA_{n}.

(b) Show that SnS_{n} is generated by the set {(12),(123)}\left\{\left(\begin{array}{lll}1 & 2\end{array}\right),\left(\begin{array}{llll}1 & 2 & 3 & \ldots\end{array}\right)\right\}.

Given 1k<n1 \leqslant k<n, prove that the set {(11+k),(123n)} generates Sn if and only \left\{\left(\begin{array}{ll}1 & \left.1+k),\left(\begin{array}{lll}1 & 2 & 3\end{array} \ldots n\right)\right\} \text { generates } S_{n} \text { if and only }\end{array}\right.\right. if kk and nn are coprime.

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