Question 1. Let G is a finite group and ø : G → G' be a surjective homomorphism (that is, ø 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 3 G, N4G, and gcd(|H|, |G/N|) = 1. Prove that H 3N. Question 3. Suppose G is a group with order 99. Prove that G must have an element of order 3. Question 4. An element g in a group G is called a square if g = a² for some a E G. Suppose that G is an Abelian 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.
Question 1. Let G is a finite group and ø : G → G' be a surjective homomorphism (that is, ø 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 3 G, N4G, and gcd(|H|, |G/N|) = 1. Prove that H 3N. Question 3. Suppose G is a group with order 99. Prove that G must have an element of order 3. Question 4. An element g in a group G is called a square if g = a² for some a E G. Suppose that G is an Abelian 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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[Groups and Symmetries] How do you solve Q1, thanks
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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