True or False Label each of the following statements as either true or false. 1. Let H be any subgroup of a group G and a Є G. Then aH = Ha implies ah = ha for all h in H. 2. The trivial subgroups {e} and G are both normal subgroups of the group G. 3. The trivial subgroups {e} and G are the only normal subgroups of a nonabelian group G. 4. Let H be a subgroup of a group G. If hH = H = Hh for all hЄ H, then H is normal in G. 5. If a group G contains a normal subgroup, then every subgroup of G must be normal. 6. Let A be a nonempty subset of a group G. Then A = (A). 7. Let A be a nonempty subset of a group G. Then (A) is closed under the group opera- tion if and only if A is closed under the same operation.
True or False Label each of the following statements as either true or false. 1. Let H be any subgroup of a group G and a Є G. Then aH = Ha implies ah = ha for all h in H. 2. The trivial subgroups {e} and G are both normal subgroups of the group G. 3. The trivial subgroups {e} and G are the only normal subgroups of a nonabelian group G. 4. Let H be a subgroup of a group G. If hH = H = Hh for all hЄ H, then H is normal in G. 5. If a group G contains a normal subgroup, then every subgroup of G must be normal. 6. Let A be a nonempty subset of a group G. Then A = (A). 7. Let A be a nonempty subset of a group G. Then (A) is closed under the group opera- tion if and only if A is closed under the same operation.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.3: Subgroups
Problem 2TFE: True or false
Label each of the following statements as either true or false, where is subgroup of...
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![True or False
Label each of the following statements as either true or false.
1. Let H be any subgroup of a group G and a Є G. Then aH = Ha implies ah = ha for
all h in H.
2. The trivial subgroups {e} and G are both normal subgroups of the
group G.
3. The trivial subgroups {e} and G are the only normal subgroups of a nonabelian group G.
4. Let H be a subgroup of a group G. If hH = H = Hh for all hЄ H, then H is normal
in G.
5. If a group G contains a normal subgroup, then every subgroup of G must be normal.
6. Let A be a nonempty subset of a group G. Then A = (A).
7. Let A be a nonempty subset of a group G. Then (A) is closed under the group opera-
tion if and only if A is closed under the same operation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e2339dd-7a22-4031-9aa9-c90187c67937%2F299d6fe2-fe1c-46bf-bd61-2458828c96ea%2F64g8bl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:True or False
Label each of the following statements as either true or false.
1. Let H be any subgroup of a group G and a Є G. Then aH = Ha implies ah = ha for
all h in H.
2. The trivial subgroups {e} and G are both normal subgroups of the
group G.
3. The trivial subgroups {e} and G are the only normal subgroups of a nonabelian group G.
4. Let H be a subgroup of a group G. If hH = H = Hh for all hЄ H, then H is normal
in G.
5. If a group G contains a normal subgroup, then every subgroup of G must be normal.
6. Let A be a nonempty subset of a group G. Then A = (A).
7. Let A be a nonempty subset of a group G. Then (A) is closed under the group opera-
tion if and only if A is closed under the same operation.
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