1. a) Consider the group (M(R), +). Is the subset H = {[0]a € R} a subgroup of (M(R),+)? If yes, prove it. If no, provide a counterexample. b) Consider the group (ZxZ, +). Is the subset K = {(x, y) | x+y> 0} a subgroup of (ZxZ,+)? If yes, prove it. If no, provide a counterexample. c) Consider the group (ZxZ, x). Is the subset D = {(x, y) | x + y = 0} a subgroup of (ZZZZ, X)? If yes, prove it. If no, provide a counterexample. NOTE: The operation here is multiplication. d) For the group (Z12, +), find all the cyclic subgroups. (ie. Find all where a Є Z12.)
1. a) Consider the group (M(R), +). Is the subset H = {[0]a € R} a subgroup of (M(R),+)? If yes, prove it. If no, provide a counterexample. b) Consider the group (ZxZ, +). Is the subset K = {(x, y) | x+y> 0} a subgroup of (ZxZ,+)? If yes, prove it. If no, provide a counterexample. c) Consider the group (ZxZ, x). Is the subset D = {(x, y) | x + y = 0} a subgroup of (ZZZZ, X)? If yes, prove it. If no, provide a counterexample. NOTE: The operation here is multiplication. d) For the group (Z12, +), find all the cyclic subgroups. (ie. Find all where a Є Z12.)
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.5: Isomorphisms
Problem 5E
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Question
![1.
a) Consider the group (M(R), +). Is the subset H = {[0]a € R} a subgroup of
(M(R),+)? If yes, prove it. If no, provide a counterexample.
b) Consider the group (ZxZ, +). Is the subset K = {(x, y) | x+y> 0} a subgroup of
(ZxZ,+)? If yes, prove it. If no, provide a counterexample.
c) Consider the group (ZxZ, x). Is the subset D = {(x, y) | x + y = 0} a subgroup of
(ZZZZ, X)? If yes, prove it. If no, provide a counterexample. NOTE: The operation
here is multiplication.
d) For the group (Z12, +), find all the cyclic subgroups. (ie. Find all <a> where
a Є Z12.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F545e4110-447f-42e0-86d8-f710095b2322%2F566fc3b9-233c-4d55-99ae-41bfffe49b4c%2Fxwpm5h9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
a) Consider the group (M(R), +). Is the subset H = {[0]a € R} a subgroup of
(M(R),+)? If yes, prove it. If no, provide a counterexample.
b) Consider the group (ZxZ, +). Is the subset K = {(x, y) | x+y> 0} a subgroup of
(ZxZ,+)? If yes, prove it. If no, provide a counterexample.
c) Consider the group (ZxZ, x). Is the subset D = {(x, y) | x + y = 0} a subgroup of
(ZZZZ, X)? If yes, prove it. If no, provide a counterexample. NOTE: The operation
here is multiplication.
d) For the group (Z12, +), find all the cyclic subgroups. (ie. Find all <a> where
a Є Z12.)
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