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Question 3. Suppose G is a group with order 99. Prove that G must have an element of order 3.

Question 3. Suppose G is a group with order 99. Prove that G must have an element of order 3.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.4: Cyclic Groups
Problem 16E: For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have...
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[Groups and Symmetries] How do you solve Q3, thanks

Question 1. Let G is a finite group and o :G → G' be a surjective homomorphism (that is, o is a homomorphism, and it is surjective). If G' has an element of order n, prove that G has an element of order n. Note. ø may not be injective. Question 2. Let G be a finite group, H < G, N 4G, and gcd(|H|,|G/N|) = 1. Prove that H < N. Question 3. Suppose G is a group with order 99. Prove that G must have an element of order 3. a? for some a E G. Suppose that G is an Abelian Question 4. An element g in a group G is called a square if g group and H is a subgroup of G. If every element of H is a square and every element of G/H is a square, prove that every element of G is a square. Question 5. (a) Prove that every element in Q/Z has finite order. (b) Prove that every non-identity element in R/Q has infinite order.
Transcribed Image Text:Question 1. Let G is a finite group and o :G → G' be a surjective homomorphism (that is, o is a homomorphism, and it is surjective). If G' has an element of order n, prove that G has an element of order n. Note. ø may not be injective. Question 2. Let G be a finite group, H < G, N 4G, and gcd(|H|,|G/N|) = 1. Prove that H < N. Question 3. Suppose G is a group with order 99. Prove that G must have an element of order 3. a? for some a E G. Suppose that G is an Abelian Question 4. An element g in a group G is called a square if g group and H is a subgroup of G. If every element of H is a square and every element of G/H is a square, prove that every element of G is a square. Question 5. (a) Prove that every element in Q/Z has finite order. (b) Prove that every non-identity element in R/Q has infinite order.
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