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Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 30E: Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.

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Let G be a group. Prove that the mapping a(g) = g-1 for all g in G
is an automorphism if and only if G is Abelian.

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Exercises 22. Let be a finite cyclic group of order with generators and . Prove that the mapping is an automorphism of .
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 G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.
15. Prove that if for all in the group , then is abelian.
41. Let be a cyclic group, . Prove that is abelian.
For a fixed group G, prove that the set of all automorphisms of G forms a group with respect to mapping composition.
Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.
26. Prove or disprove that if a group has an abelian quotient group , then must be abelian.
Suppose that G and G are abelian groups such that G=H1H2 and G=H1H2. If H1 is isomorphic to H1 and H2 is isomorphic to H2, prove that G is isomorphic to G.
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.
Let G be an abelian group. Prove that the set of all elements of finite order in G forms a subgroup of G. This subgroup is called the torsion subgroup of G.
Suppose that G and H are isomorphic groups. Prove that G is abelian if and only if H is abelian.

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