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Prove that for any non-trivial subgroups A and B of Q (this is the group of rational numbers with operation +), the intersection An B is non-trivial Explain why this proves that Q cannot be isomorphic to any direct product of non-trivia groups.

Prove that for any non-trivial subgroups A and B of Q (this is the group of rational numbers with operation +), the intersection An B is non-trivial Explain why this proves that Q cannot be isomorphic to any direct product of non-trivia groups.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.7: Direct Sums (optional)
Problem 19E: 19. a. Show that is isomorphic to , where the group operation in each of , and is addition. ...
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Prove that for any non-trivial subgroups A and B of Q (this is the group of rational numbers with operation +), the intersection AB is non-trivial. Explain why this proves that Q cannot be isomorphic to any direct product of non-trivial groups.
Transcribed Image Text:Prove that for any non-trivial subgroups A and B of Q (this is the group of rational numbers with operation +), the intersection AB is non-trivial. Explain why this proves that Q cannot be isomorphic to any direct product of non-trivial groups.
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