Suppose that o is an isomorphism from a group G onto a group G. Then 1. 6-1 is an isomorphism from G onto G. 2. G is Abelian if and only if G is Abelian. 3. G is cyclic if and only if G is cyclic. 4. If K is a subgroup of G, then 4(K) = {$(k) I k E K} is a subgroup of G. 5. If K is a subgroup of G, then o–1 (K) = {g E G I ¢(g) E K} is a subgroup of G. 6. 4(Z(G)) = Z(G).

Suppose that o is an isomorphism from a group G onto a group G. Then 1. 6-1 is an isomorphism from G onto G. 2. G is Abelian if and only if G is Abelian. 3. G is cyclic if and only if G is cyclic. 4. If K is a subgroup of G, then 4(K) = {$(k) I k E K} is a subgroup of G. 5. If K is a subgroup of G, then o–1 (K) = {g E G I ¢(g) E K} is a subgroup of G. 6. 4(Z(G)) = Z(G).

Name: Prove property 4 Suppose that o is an isomorphism from a group G onto
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Description: Prove property 4 Suppose that o is an isomorphism from a group G onto a group G.Then1. 6-1 is an isomorphism from G onto G.2. G is Abelian if and only if G is Abelian.3. G is cyclic if and only if G is cyclic.4. If K is a subgroup of G, then 4(K) = {

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.7: Direct Sums (optional)

Problem 10E: 10. Suppose that and are subgroups of the abelian group such that . If is a subgroup of such...

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