18. An external direct product G₁ G₂ G₂of a finite number of cyclic groups is cyclic if and only if |G₁ and |G₁| are relatively prime when i + j. TRUE FALSE 19. The group ZmZn is cyclic and isomorphic to Zmn if and only if the least common multiple of m and n is 1. TRUE FALSE 20. ZmZn has mn elements whether m and n are relatively prime or not. ● TRUE
18. An external direct product G₁ G₂ G₂of a finite number of cyclic groups is cyclic if and only if |G₁ and |G₁| are relatively prime when i + j. TRUE FALSE 19. The group ZmZn is cyclic and isomorphic to Zmn if and only if the least common multiple of m and n is 1. TRUE FALSE 20. ZmZn has mn elements whether m and n are relatively prime or not. ● TRUE
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.4: Cyclic Groups
Problem 39E
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Please answer question 18, 19 and 20. Answer if TRUE or FALSE only.
![18. An external direct product G₁ G₂ G₂of a finite number of cyclic groups is cyclic
if and only if |G₁| and |G₁| are relatively prime when i ‡ j.
TRUE
FALSE
19.
The group ZmZn is cyclic and isomorphic to Zmn if and only if the least common
multiple of m and n is 1.
TRUE
FALSE
20.
ZmZn has mn elements whether m and n are relatively prime or not.
●
TRUE
FALSE](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e69f71c-493f-4e8c-97ca-6b833368d1d8%2F5e1da3a0-1742-4aeb-8238-a4c96e388313%2Fl7pc284_processed.jpeg&w=3840&q=75)
Transcribed Image Text:18. An external direct product G₁ G₂ G₂of a finite number of cyclic groups is cyclic
if and only if |G₁| and |G₁| are relatively prime when i ‡ j.
TRUE
FALSE
19.
The group ZmZn is cyclic and isomorphic to Zmn if and only if the least common
multiple of m and n is 1.
TRUE
FALSE
20.
ZmZn has mn elements whether m and n are relatively prime or not.
●
TRUE
FALSE
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