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Problem 3. Suppose G is a group. (i) Show that the following are equivalent: (1) For all n E Z, the function p(g) = g" is a homomorphism from G to G. (2) The function p(g) = g? is a homomorphism from G to G. (3) G is abelian. (ii) An element g E G is an n-th power if g = h" for some he G. Show that if G is a then the number of n-th powers in G is equal to the cardinality of G divided by the number of elements g E G such that the order of g divides n. finite abelian group,

Problem 3. Suppose G is a group. (i) Show that the following are equivalent: (1) For all n E Z, the function p(g) = g" is a homomorphism from G to G. (2) The function p(g) = g? is a homomorphism from G to G. (3) G is abelian. (ii) An element g E G is an n-th power if g = h" for some he G. Show that if G is a then the number of n-th powers in G is equal to the cardinality of G divided by the number of elements g E G such that the order of g divides n. finite abelian group,

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.2: Properties Of Group Elements
Problem 23E: 23. Let be a group that has even order. Prove that there exists at least one element such that and...
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Problem 3. Suppose G is a group. (i) Show that the following are equivalent: (1) For all n E Z, the function p(g) = g" is a homomorphism from G to G. (2) The function p(g) = g² is a homomorphism from G to G. (3) G is abelian. (ii) An element g E G is an n-th power if g finite abelian group, then the number of n-th powers in G is equal to the cardinality of G divided by the number of elements g EG such that the order of g divides n. h" for some he G. Show that if G is a
Transcribed Image Text:Problem 3. Suppose G is a group. (i) Show that the following are equivalent: (1) For all n E Z, the function p(g) = g" is a homomorphism from G to G. (2) The function p(g) = g² is a homomorphism from G to G. (3) G is abelian. (ii) An element g E G is an n-th power if g finite abelian group, then the number of n-th powers in G is equal to the cardinality of G divided by the number of elements g EG such that the order of g divides n. h" for some he G. Show that if G is a
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