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Let H = (a) be a cyclic group. Define f: Z → H by f(k) = a*. Prove that f is a surjective homomorphism. (a) Suppose |a| = n < x. Prove that ker f = nZ. Prove that the First Isomorphism Theorem implies H= Zn. (b) If a = ∞o, prove that HZ.

Let H = (a) be a cyclic group. Define f: Z → H by f(k) = a*. Prove that f is a surjective homomorphism. (a) Suppose |a| = n < x. Prove that ker f = nZ. Prove that the First Isomorphism Theorem implies H= Zn. (b) If a = ∞o, prove that HZ.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.6: Quotient Groups
Problem 23E
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Let H = (a) be a cyclic group. Define f: Z → H by f(k) = a*. Prove that f is a surjective homomorphism. (a) Suppose |a| = n < x. Prove that ker f = nZ. Prove that the First Isomorphism Theorem implies HZ. (b) If a ∞o, prove that HZ. =
Transcribed Image Text:Let H = (a) be a cyclic group. Define f: Z → H by f(k) = a*. Prove that f is a surjective homomorphism. (a) Suppose |a| = n < x. Prove that ker f = nZ. Prove that the First Isomorphism Theorem implies HZ. (b) If a ∞o, prove that HZ. =
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