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**Question**

**Let**[itex]S_n[/itex]

**be the symmetric group on**[itex]n[/itex]

**letters.**

(i)

**Show that if**[itex]\sigma = (x_1,\dots,x_k)[/itex]

**is a cycle and**[itex]\phi \in S_n[/itex]

**then**

[tex]\phi\sigma\phi^{-1} = (\phi(x_1),\dots,\phi(x_k))[/tex]

(ii)

**Show that the congujacy class of a permutation [itex]\sigma \in S_n[/itex] consists of all permutations in [itex]S_n[/itex] of the same cycle type as [itex]\sigma[/itex]**

(iii)

**In the case of [itex]S_5[/itex], give the numbers of permutations of each cycle type**

(iv)

**Find all normal subgroups of [itex]S_5[/itex]**