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6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and only if G' is abelian. (b). o(a) = o(y(a)). (c). G is cyclic if and only if G' is cyclic.

6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and only if G' is abelian. (b). o(a) = o(y(a)). (c). G is cyclic if and only if G' is cyclic.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.5: Isomorphisms
Problem 33E: Suppose that G and H are isomorphic groups. Prove that G is abelian if and only if H is abelian.
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6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and only if G" is abelian. (b). o(a) = o(p(a)). (c). G is cyclic if and only if G' is cyclic.
Transcribed Image Text:6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and only if G" is abelian. (b). o(a) = o(p(a)). (c). G is cyclic if and only if G' is cyclic.
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