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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).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove property 4

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

We have to prove ϕK=ϕk|k∈K is a subgroup of G.

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