5. True or False. Explain your answers. (a) If every element of G/H has finite order, and every element of H has finite order, then every element of G has finite order. (b) If |G/H|= =p, then the order of every element a H in G is a multiple of p.
5. True or False. Explain your answers. (a) If every element of G/H has finite order, and every element of H has finite order, then every element of G has finite order. (b) If |G/H|= =p, then the order of every element a H in G is a multiple of p.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question

Transcribed Image Text:5. True or False. Explain your answers.
(a) If every element of G/H has finite order, and every element of H has finite order, then every element of G
has finite order.
(b) If |G/H| = p, then the order of every element a & H in G is a multiple of p.
Hint: If a” = e for some integer n not divisible by p, then n and p are relatively prime and there exist integers î
and y such that nx+py = 1. Consider anx+py and get a contradiction.
Expert Solution

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Step 1: In this step, we will provide the explanation of the first True or False question.
VIEWStep 2: In this step, we will show that xH=H if and only if x is in H.
VIEWStep 3: Finally, we have shown that the order of every element of G is of finite order
VIEWStep 4: In this step, we will provide the solution of the second question.
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