Find the non-isomorphic abelian groups of order 360.
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Find the non-isomorphic abelian groups of
order 360.
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Solved in 3 steps with 1 images
- 25. Prove or disprove that every group of order is abelian.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 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.
- Exercises 35. Prove that any two groups of order are isomorphic.27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.15. Prove that if for all in the group , then is abelian.
- Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.9. Find all homomorphic images of the octic group.