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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
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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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.
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.
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