let f: G1->G2 be an isomorphism then show that G1 is cyclic iff G2 is cyclic
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Originally Posted by mathemanyak let f: G1->G2 be an isomorphism then show that G1 is cyclic iff G2 is cyclic is cyclic iff there exists such that . Okay, we now say, first, since there's a homomorphism: (1) Since it must be because (the order of a is |G_1|) being all those elements different. is biyective thus: by (1): thus The converse is analogous, take
Originally Posted by PaulRS 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
Last edited by JaneBennet; August 3rd 2008 at 10:06 AM.
JaneBennet is right but, considering the fact that PaulRS is only 18 years old, it's fascinating that how knowledgeable he is!
Originally Posted by JaneBennet Thus, for the first part: We have that so we only need to show that . for some This shows that only needs to be a surjective homomorphism in order for cyclic cyclic to hold.
Originally Posted by algebraic topology This shows that only needs to be a surjective homomorphism in order for cyclic cyclic to hold. Hence, if is an isomorphism, the fact that is a surjective homomorphism from onto proves the reverse implication and completes the proof.
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