Let phi be a homomorphism from Z/30Z to Z/5Z with phi(6) = 2. Determine the image of phi.
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Let phi be a homomorphism from Z/30Z to Z/5Z with phi(6) = 2. Determine the image of phi.
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- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.Let be as described in the proof of Theorem. Give a specific example of a positive element of .Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
- 6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.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.Let be a subgroup of a group with . Prove that if and only if .
