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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.

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
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.3: Subgroups
Problem 9TFE
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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.
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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