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- Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.Let be as described in the proof of Theorem. Give a specific example of a positive element of .
- Prove that Cay({(0, 1), (1, 1)}:Zm ⊕Zn) has a Hamiltonian circuitfor all m and n greater than 1.Compute the Wronskian for 2t^2y''+3ty'-y=01-Consider the set S3, the group of permutations on three elements {1, 2, 3}. Find two elements a and b in S3such that (ab)2 not equal to a2b2 and hence conclude from the previous question that S3 is not abelian. plz with details