3. Let m and n be positive integers such that m divides n. (a) Prove that the map y: Z/nZ Z/mZ sending a + nZ to a + mZ for any a E Z is well-defined. (b) Prove that y is a surjective group homomorphism. (c) Determine the group structure of the kernel of p.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Why or why not?
3. Let m and n be positive integers such that m divides n.
(a) Prove that the map p: Z/nZ Z/mZ sending a + nZ to a + mZ for any a e Z is
well-defined.
(b) Prove that y is a surjective group homomorphism.
(c) Determine the group structure of the kernel of p.
Transcribed Image Text:Why or why not? 3. Let m and n be positive integers such that m divides n. (a) Prove that the map p: Z/nZ Z/mZ sending a + nZ to a + mZ for any a e Z is well-defined. (b) Prove that y is a surjective group homomorphism. (c) Determine the group structure of the kernel of p.
Expert Solution
part a

Given m and n are two positive integers such that m divides n.

To prove the map φ:/n/m such that φa+n=a+m for any a is well-defined

Let us consider two integers a and a' such that

 a+n=a'+n                       (i)

Ten, eq. (i) implies n divides a-a'. Now, according to the definition of φ

φa+n=a+m

φa'+n=a'+m

Since n divides a-a' and m divides n, by transitivity, m divides a-a'. So a+m=a'+m implying 

φa+n=φa'+n

Hence φ is well-defined

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