(3) Suppose N is a normal subgroup of G such that N [G, G] is the trivial subgroup. Show that N is contained in the center Z of G.
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- Find two groups of order 6 that are not isomorphic.32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping defined by is an automorphism of . Each of these automorphism is called an inner automorphism of . Prove that the set forms a normal subgroup of the group of all automorphism of . Exercise 20 of Section 3.5 20. For each in the group , define a mapping by . Prove that is an automorphism of .Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.
- Let H be a torsion subgroup of an abelian group G. That is, H is the set of all elements of finite order in G. Prove that H is normal in G.Write 20 as the direct sum of two of its nontrivial subgroups.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.
- 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.Exercise 8 states that every subgroup of an abelian group is normal. Give an example of a nonabelian group for which every subgroup is normal. Exercise 8: Show that every subgroup of an abelian group is normal.15. Prove that if for all in the group , then is abelian.
- 25. Prove or disprove that every group of order is abelian.Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.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.