IIT-JAM 2018 Abstract Algebra Question (3)

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General Mathematics

Question
Solution: we know that, 

Number of homomorphisms from \mathbb{Z}_n to \mathbb{Z}_m is gcd(m,n)

Now, as 221=13\times 17. Hence, gcd(51,221)≠1, gcd(91,221)≠1, gcd(119,221)≠1, But gcd(21,221)=1

Hence number of homomorphisms from \mathbb{Z}_{21} to \mathbb{Z}_{221} is 1 that is, there is only one homomorphism from \mathbb{Z}_{21} to \mathbb{Z}_{221}. So that this must be trivial homomorphism( zero map). 

Hence if f:\mathbb{Z}_{21}\rightarrow\mathbb{Z}_{221} is homomorphism then it is trivial homomorphism ( zero map) i.e. f(a)=0 for all a\in\mathbb{Z}_{21}.

Hence option (A) is correct. 

(Note other options are not correct because in those case there exists non-trivial homomorphisms, because gcd of numbers 51,91,119 with 221 is not 1.)

Hence answer is "option (A)" only.

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