True or False Questions for Chapter 10

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True or False Questions for Chapter 10 - 11

1. Every isomorphism is a homomorphism.

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True

False

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2. Every homomorphism is an isomorphism.

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True

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3. There is at least one homomorphism from any group G to any group G'.

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True

False

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4. The homomorphic image of a cyclic group is cyclic.

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True

False

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5. The homomorphic image of an Abelian group is Abelian.

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True

False

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6. A nontrivial homomorphic image of a non-Abelian group is non-Abelian.

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True

False

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7. If f is a homomorphism then |f(a)| divides |a|.

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True

False

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8. If f is a homomorphism from G to K and H is normal in G then f(H) is normal in K.

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True

False

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9. If f is a homomorphism from G to a group and H is finite subgroup of G, then |f(H)| divides |H|.

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True

False

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10. If the homomorphic image of a group G has an element of order n then G has an element of order n.

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True

False

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11. Every homomorphic image of a group G is isomorphic to a factor group of G.

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True

False

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12. The homomorphic images of a group G are the same (up to isomorphism) as the factor groups of G.

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True

False

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13. For every divisor m of the order of a finite Abelian group there is a subgroup of order m.

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True

False

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