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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 24E: 24. Let be a cyclic group. Prove that for every normal subgroup of , is a cyclic group.

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Find an example of a group G acting transitively
on a set X (G has only one orbit) and a subgroup
H of G which has less than [G:H]=n orbits.

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9. Suppose that and are subgroups of the abelian group such that . Prove that .
Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?
Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.
Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.
Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.
18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.
Exercises 35. Prove that any two groups of order are isomorphic.
Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.
Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .
Label each of the following statements as either true or false. Two groups can be isomorphic even though their group operations are different.

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