derive all groups of order 12 using sylow theorems. please dont use any generalizations show all...

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derive all groups of order 12 using sylow theorems. please dont useany generalizations show all work and theorems used. all 5 ofthem

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will appeal to an isomorphism property of semidirect products for any semidirect product H o K and automorphism f K K H o K H of K This says that precomposing an action of K on H by automorphisms thats with an automorphism of K produces an isomorphic semidirect product of H and K Proof Let G 12 22 3 A 2Sylow subgroup has order 4 and a 3Sylow subgroup has order 3 We will start by showing G has a normal 2Sylow or 3Sylow subgroup n2 1 or n3 1 From the Sylow theorems n2 3 n2 1 mod 2 n3 4 n3 1 mod 3 Therefore n2 1 or 3 and n3 1 or 4 To show n2 1 or n3 1 assume n3 6 1 Then n3 4 Lets count elements of order 3 Since each 3Sylow subgroup has prime order 3 two different 3Sylow subgroups intersect trivially Each of the four 3Sylow subgroups of G contains two elements of order 3 and these are not in any other 3Sylow subgroup so the number of elements in G of order 3 is 2n2 8 This leaves us with 12 8 4 elements in G not of order 3 A 2Sylow subgroup has order 4 and contains no elements of order 3 so one 2Sylow subgroup must account for the remaining 4 elements of G Thus n2 1 if n3 6 1 Next we show G is a semidirect product of a 2Sylow and 3Sylow subgroup Let P be a 2Sylow subgroup and Q be a 3Sylow subgroup of G Since P and Q have relatively prime orders P Q 1 and the set P Q xy x P y Q has size PQP Q 12 G so G P Q Since P or Q is normal in G G is a See Answer

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