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Show a group is a semi direct product
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[QUOTE="andrewkirk, post: 5252503, member: 265790"] The latex code for semidirect product is \rtimes. The problem's notation is muddled. It should ask you to prove either (1) [I]isomorphism[/I], not equality, ie to prove that ##S_n\cong \mathbb{Z}_2\rtimes Alt(n)##, where the semidirect product is [B]Outer[/B]. OR (2) [I]equality[/I], where the semidirect product is [B]Inner[/B], ie ##S_n=N\rtimes Alt(n)## where ##N## is any subgroup of ##S_n## of order 2, which hence must be isomorphic to ##\mathbb{Z}_2##. I suggest trying for the second one. The notation is valid. If ##G,H## are subgroups ##GH## is defined as the set of all elements that can be written as ##gh## where ##g\in G,\ h\in H##. That is not necessarily a subgroup. So part of what you have to show is that ##S_n= \mathbb{Z}_2 Alt(n)## (or rather ##N\,Alt(n)## using my notation of (2) above) is a subgroup. Why not pick N to be the subgroup generated by the swap permutation (1 2). Then try to prove the two things you need to prove. The intersection one is dead easy. The other, not so much. [/QUOTE]
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