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10 Nov 2019
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Prove that it is a homomorphism
Let k and l be fixed integers. Define k.l : Z Z by = k,1(m,n) = km + ln, m,n Z. Prove that k,l is a homomorphism. Let if = {(2m, - m) | m 6 Z }. Prove that H is a subgroup of Z Z. Use the First Isomorphism Theorem to prove that the factor group (Z Z)/H is isomorphic to Z. Remark: You were asked to use the First Isomorphism Theorem to solve part b). There is an alternate way to solve part b), by showing that the factor group (Z Z)/H is an infinite cyclic group. An infinite cyclic group is isomorphic to Z.
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Prove that it is a homomorphism
Let k and l be fixed integers. Define k.l : Z Z by = k,1(m,n) = km + ln, m,n Z. Prove that k,l is a homomorphism. Let if = {(2m, - m) | m 6 Z }. Prove that H is a subgroup of Z Z. Use the First Isomorphism Theorem to prove that the factor group (Z Z)/H is isomorphic to Z. Remark: You were asked to use the First Isomorphism Theorem to solve part b). There is an alternate way to solve part b), by showing that the factor group (Z Z)/H is an infinite cyclic group. An infinite cyclic group is isomorphic to Z.
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Nelly StrackeLv2
13 May 2019