Homework Answers
Let D4 be dihedral group order 8. So D4={e, a, a^2, a^3, b, ab, a^2b, a^3b}, a^4 = e,...
Let D4 be dihedral group order 8. So D4={e, a, a^2, a^3, b, ab, a^2b, a^3b}, a^4 = e, b^2= e, ab=ba^3; A. FIND ALL THE COSETS OF THE SUBGROUP H= , list their elements. B. What is the index [D4 : H] C. DETERMINE IF H IS NORMAL
Problem 3.[10 points.] Let D. be the dihedral group of order 2n. Let H = (r)...
Problem 3.[10 points.] Let D. be the dihedral group of order 2n. Let H = (r) be the subgroup of D, consisting of all rotations. Prove that every subgroup of H is normal in D Solution:
Let a : G + H be a homomorphism. Which of the following statements must necessarily...
Let a : G + H be a homomorphism. Which of the following statements must necessarily be true? Check ALL answers that are necessarily true. There may be more than one correct answer. A. If kera is trivial (i.e., ker a = {eg}), then a is injective. B. If the image of a equals H, then a is injective. C. The first isomorphism theorem gives an isomorphism between the image of a and a certain quotient group. D. The first...
(7 marks) Let n be a positive integer and let G be a group such that...
(7 marks) Let n be a positive integer and let G be a group such that there is a surjective homomorphism from G onto the symmetric group Sn. Show that G has a normal subgroup of index 2.
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...
Can you explain these for me..I mean give example for each of them to get the...
Can you explain these for me..I mean give example for
each of them to get the idea
THE FIRST ISOMORPHISM THEOREM Let: G M be a homomorphism with Ker() = K, and Im() = I. Then there is a natural isomorphism 0:1 ™G/K which is surjective. Thus, I G/K. THE SECOND ISOMORPHISM THEOREM Suppose the N is a normal subgroup of G, and that H is a subgroup of G. Then H/( HN) = (HN)/N. THE THIRD ISOMORPHISM THEOREM Let...
18. Let N be a normal subgroup of a finite group G, and let Nxi, ....
18. Let N be a normal subgroup of a finite group G, and let Nxi, . N be a for complete list of (disjoint) right cosets. Prove that, as sets, Nx, Nz all i and j Nz,
2. Let n 2 3, and G D2n e,r,r2,... ,r"-1,s, sr, sr2,..., sr-'), the dihedral group with 2n ele- 3...
2. Let n 2 3, and G D2n e,r,r2,... ,r"-1,s, sr, sr2,..., sr-'), the dihedral group with 2n ele- 3, ST, ST,..,ST ments. We let R-(r) denote the subgroup consisting of all rotations. (a) Show that, if M is a subgroup of R, and is in GR, then the union M UrM is a subgroup of G. Here xM-{rm with m in M) (b) Now take n- 12 and M (). How many distinct subgroups does the construction in (a)...
Let G be a group of order 16, such that each element can be written (uniquely) in the form rasb, ...
Let G be a group of order 16, such that each element can be written (uniquely) in the form rasb, where a є {0, ,7} and b є {0,1). The elements r and s satisfy the relations: r"=1; s2 = 1; sr=r38. (The final relation means that an s can be moved past an r if we raise the r to the third power.) Let H = {1,s). Let . : G × G/H → G/H be the usual action...
5. Let N be a normal subgroup of a group G and G/N be the quotient...
5. Let N be a normal subgroup of a group G and G/N be the quotient group of all right cosets of N in G. Prove each of the following: (a) (2 pts) If G is cyclic, then so is G/N. (b) (3 pts) G/N is Abelian if and only if aba-16-? E N Va, b E G. (c) (3 pts) If G is a finite group, then o(Na) in G/N is a divisor of (a) VA EG.
Problem 3.[10 points.] Let D. be the dihedral group of order 2n. Let H = (r) be the subgroup of D, consisting of all rotations. Prove that every subgroup of H is normal in D Solution:
Let a : G + H be a homomorphism. Which of the following statements must necessarily be true? Check ALL answers that are necessarily true. There may be more than one correct answer. A. If kera is trivial (i.e., ker a = {eg}), then a is injective. B. If the image of a equals H, then a is injective. C. The first isomorphism theorem gives an isomorphism between the image of a and a certain quotient group. D. The first...
(7 marks) Let n be a positive integer and let G be a group such that there is a surjective homomorphism from G onto the symmetric group Sn. Show that G has a normal subgroup of index 2.
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...
Can you explain these for me..I mean give example for
each of them to get the idea
THE FIRST ISOMORPHISM THEOREM Let: G M be a homomorphism with Ker() = K, and Im() = I. Then there is a natural isomorphism 0:1 ™G/K which is surjective. Thus, I G/K. THE SECOND ISOMORPHISM THEOREM Suppose the N is a normal subgroup of G, and that H is a subgroup of G. Then H/( HN) = (HN)/N. THE THIRD ISOMORPHISM THEOREM Let...
18. Let N be a normal subgroup of a finite group G, and let Nxi, . N be a for complete list of (disjoint) right cosets. Prove that, as sets, Nx, Nz all i and j Nz,
2. Let n 2 3, and G D2n e,r,r2,... ,r"-1,s, sr, sr2,..., sr-'), the dihedral group with 2n ele- 3, ST, ST,..,ST ments. We let R-(r) denote the subgroup consisting of all rotations. (a) Show that, if M is a subgroup of R, and is in GR, then the union M UrM is a subgroup of G. Here xM-{rm with m in M) (b) Now take n- 12 and M (). How many distinct subgroups does the construction in (a)...
Let G be a group of order 16, such that each element can be written (uniquely) in the form rasb, where a є {0, ,7} and b є {0,1). The elements r and s satisfy the relations: r"=1; s2 = 1; sr=r38. (The final relation means that an s can be moved past an r if we raise the r to the third power.) Let H = {1,s). Let . : G × G/H → G/H be the usual action...
5. Let N be a normal subgroup of a group G and G/N be the quotient group of all right cosets of N in G. Prove each of the following: (a) (2 pts) If G is cyclic, then so is G/N. (b) (3 pts) G/N is Abelian if and only if aba-16-? E N Va, b E G. (c) (3 pts) If G is a finite group, then o(Na) in G/N is a divisor of (a) VA EG.
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