let f: G1->G2 be an isomorphism then show that G1 is cyclic iff G2 is cyclic

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- August 3rd 2008, 05:22 AMmathemanyakiso and cyclic
let f: G1->G2 be an isomorphism then show that G1 is cyclic iff G2 is cyclic

- August 3rd 2008, 05:58 AMPaulRS
- August 3rd 2008, 08:25 AMJaneBennet
You are assuming that is finite. The question doesn’t say anything about finite groups, though.

If is cyclic and is an isomorphism, you just have to show that . Similarly, if is cyclic, then .

Thus, for the first part:

We have that so we only need to show that .

for some

- August 3rd 2008, 08:58 AMNonCommAlg
**JaneBennet**is right but, considering the fact that**PaulRS**is only 18 years old, it's fascinating that how knowledgeable he is! (Clapping) - August 3rd 2008, 05:36 PMalgebraic topology
- August 5th 2008, 08:29 AMJaneBennet