Homework Answers
1. Give an example of a group, G, and a proper subgroup, H, where H has...
1. Give an example of a group, G, and a proper subgroup, H, where H has finite index in G and H has infinite order 2. Give an example of a group, G, and a proper subgroup, H, where H has infinite index in G and H has finite order. (Hint: you won't be able to find this with the groups that we work a lot with. Try looking in SO2(R))
1. Give an example of a group, G, and...
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi...
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi-1 Gi means Gi-1 is a normal subgroup of Gi. For example, any abelian group is solvable: If G s abelian, take Go f1), Gi- G. Then G1/Go G is abelian and hence G is solvable (a) Show that S3 is solvable Suggestion: Let Go- [l),Gı-(123)), and G2 -G. Here (123)) is the subgroup generated by the 3-cycle...
4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G 4 (a) Let G be...
4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G
4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G
(5 points) Recall the Definition: A subgroup H of G is called a normal subgroup of...
(5 points) Recall the Definition: A subgroup H of G is called a normal subgroup of G if gH = Hg for all g E G. If so, we write H G. Mark each of the following true T or false F (using the CAPITAL LETTER T or F. Recall that if a statment is not necessarily ALWAYS true, then it is false. - T ח 1. Every subgroup of (Zn, e) is normal. 2. The cyclic group (f) is...
22 Must the center of a group be Abelian? 23. Let G be an Abelian group...
22 Must the center of a group be Abelian? 23. Let G be an Abelian group with identity e and let n be some integer Prove that the set of all élements of G that satisfy the equation* - e is a subgroup of G. Give an example of a group G in which the set of all elements of G that satisfy the equation :2 -e does not form a subgroup of G. (This exercise is referred to in...
(a) Show that if H and K are subgroups of an abelian group G, then HK...
(a) Show that if H and K are subgroups of an abelian group G, then HK = {hk|he H, k E K} is a subgroup of G (b) Show that if H and K are normal subgroups of a group G, then H N K is a normal subgroup of G
abstract algebra show your work 3. Let H be a subgroup of G with |G|/\H =...
abstract algebra
show your work
3. Let H be a subgroup of G with |G|/\H = 2. Prove that H is normal in G. Hint: Let G. If Hthen explain why xH is the set of all elements in G not in H. Is the same true for H.C? Remark: The above problem shows that A, is a normal subgroup of the symmetric group S, since S/A, 1 = 2. It also shows that the subgroup Rot of all rotations...
Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a...
Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a normal subgroup. Let K be a subgroup of G that contains H. (1) Show that there is a well-defined injective homomorphism i: K/ H G /H given by i(kH) = kH. By abuse of notation, we regard K/H as being the subgroup Imi < G/H consisting of all cosets of the form KH with k EK. (2) Show that every subgroup of G/H is...
(4)(20 points) (a) Show that if H and K are subgroups of an abelian group G,...
(4)(20 points) (a) Show that if H and K are subgroups of an abelian group G, then HK = {hk|he H, KE K} is a subgroup of G. (b) Show that if H and K are normal subgroups of a group G, then HK is a normal subgroup of G
(4)(20 points) (a) Show that if H and K are subgroups of an abelian group G,...
(4)(20 points) (a) Show that if H and K are subgroups of an abelian group G, then HK = {hk|he H, ke K}is a subgroup of G (b) Show that if Hand K are normal subgroups of a group G, then H N K is a normal subgroup of G
1. Give an example of a group, G, and a proper subgroup, H, where H has finite index in G and H has infinite order 2. Give an example of a group, G, and a proper subgroup, H, where H has infinite index in G and H has finite order. (Hint: you won't be able to find this with the groups that we work a lot with. Try looking in SO2(R))
1. Give an example of a group, G, and...
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi-1 Gi means Gi-1 is a normal subgroup of Gi. For example, any abelian group is solvable: If G s abelian, take Go f1), Gi- G. Then G1/Go G is abelian and hence G is solvable (a) Show that S3 is solvable Suggestion: Let Go- [l),Gı-(123)), and G2 -G. Here (123)) is the subgroup generated by the 3-cycle...
4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G
4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G
(5 points) Recall the Definition: A subgroup H of G is called a normal subgroup of G if gH = Hg for all g E G. If so, we write H G. Mark each of the following true T or false F (using the CAPITAL LETTER T or F. Recall that if a statment is not necessarily ALWAYS true, then it is false. - T ח 1. Every subgroup of (Zn, e) is normal. 2. The cyclic group (f) is...
22 Must the center of a group be Abelian? 23. Let G be an Abelian group with identity e and let n be some integer Prove that the set of all élements of G that satisfy the equation* - e is a subgroup of G. Give an example of a group G in which the set of all elements of G that satisfy the equation :2 -e does not form a subgroup of G. (This exercise is referred to in...
(a) Show that if H and K are subgroups of an abelian group G, then HK = {hk|he H, k E K} is a subgroup of G (b) Show that if H and K are normal subgroups of a group G, then H N K is a normal subgroup of G
abstract algebra
show your work
3. Let H be a subgroup of G with |G|/\H = 2. Prove that H is normal in G. Hint: Let G. If Hthen explain why xH is the set of all elements in G not in H. Is the same true for H.C? Remark: The above problem shows that A, is a normal subgroup of the symmetric group S, since S/A, 1 = 2. It also shows that the subgroup Rot of all rotations...
Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a normal subgroup. Let K be a subgroup of G that contains H. (1) Show that there is a well-defined injective homomorphism i: K/ H G /H given by i(kH) = kH. By abuse of notation, we regard K/H as being the subgroup Imi < G/H consisting of all cosets of the form KH with k EK. (2) Show that every subgroup of G/H is...
(4)(20 points) (a) Show that if H and K are subgroups of an abelian group G, then HK = {hk|he H, KE K} is a subgroup of G. (b) Show that if H and K are normal subgroups of a group G, then HK is a normal subgroup of G
(4)(20 points) (a) Show that if H and K are subgroups of an abelian group G, then HK = {hk|he H, ke K}is a subgroup of G (b) Show that if Hand K are normal subgroups of a group G, then H N K is a normal subgroup of G
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