- #1
Oxymoron
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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]
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]