Show that the general linear group GL(2,Z2) is isomorphic to the symmetric group S3. (Hint. Write...
Show that the general linear group GL(2,Z2) is isomorphic to the symmetric group S3. (Hint. Write out the multiplication tables for both groups
Homework Answers
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Show that the general linear group GL(2,Z2) is isomorphic to the
symmetric group S3. (Hint. Write...
(h) Show that the affine group AGL(1,q) is isomorphic to a subgroup of GL(2,9), the general...
(h) Show that the affine group AGL(1,q) is isomorphic to a subgroup of GL(2,9), the general linear group of non-singular matrices over GF(q), by using the mapping ax + b (Why is this an isomorphism?) [10] (8 h
(2) Consider the following groups: Z24; Z3 x Z7 x Z2; Z2 x Z2 x S3; Zg x Z3; G-symmetries of the squareZ. Which of these groups are isomorphic to one another? (2) Consider the following groups:...
(2) Consider the following groups: Z24; Z3 x Z7 x Z2; Z2 x Z2 x S3; Zg x Z3; G-symmetries of the squareZ. Which of these groups are isomorphic to one another?
(2) Consider the following groups: Z24; Z3 x Z7 x Z2; Z2 x Z2 x S3; Zg x Z3; G-symmetries of the squareZ. Which of these groups are isomorphic to one another?
1. Show that S3 is not isomorphic to U(14) 2. How many automorphisms are there from...
1. Show that S3 is not isomorphic to U(14) 2. How many automorphisms are there from Z2 to itself?Define each of them.
How many non-isomorphic unital rings are there of order 4? Question 3: How many non-isomorphic unital...
How many non-isomorphic unital rings are there of order 4?
Question 3: How many non-isomorphic unital rings R4 are there of order 4? Hint: we can assume that the additive group of R4 can be either (74, +) or (Z2 X Z2, +). Thus the elements of R4 are one or the other of these groups, with a multiplication defined in some way. In the former case, 1 can be assumed to be the multiplicative identity. Why can't 2 be...
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the ...
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the argument used in lecture to show that R[z]/(x2 1) is isomorphic to C)
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the argument used in lecture to show that R[z]/(x2 1) is isomorphic to C)
8. (10 points) Consider the general linear group GL(2,26) = {© a) | where a, b,...
8. (10 points) Consider the general linear group GL(2,26) = {© a) | where a, b, c, d e Zes and ad – be + 0} (a) Determine the order of the group GL(2, Z5). (b) List a Sylow 5-subgroup of GL(2, Z5).
Problem 3. Consider the general linear group GL2 = (M2,*) of 2 x 2 invertible matrices...
Problem 3. Consider the general linear group GL2 = (M2,*) of 2 x 2 invertible matrices under matrix multiplication. In Homework Problem 9 of Investigation 6, you showed that Pow G 1-( )z is isomorphic to the group Z. Prove that the group (Pow 1 i
problem 4a in worksheet 2 11. Recall from problem 4a on Algebra Problem Sheet 2 that the general linear group...
problem 4a in worksheet 2
11. Recall from problem 4a on Algebra Problem Sheet 2 that the general linear group GL2(R) is the set of 2 x 2 matrices ahwhere a, b,c,d are real numbers such that ad be 0 under matrix multiplication, which is defined by (a) Prove that the set H-( [劙 adメ0} is a subgroup of GL2(R). (b) Let A = 1] and B-| 의 히 . Show that ord (A)-3, ord (B) = , and ord...
11. Let G = Z4 Z4, H = {0,0), (2,0), (0,2), (2,2)). Write the Cayley table...
11. Let G = Z4 Z4, H = {0,0), (2,0), (0,2), (2,2)). Write the Cayley table for G/H. Is G/H isomorphic to Z4 or Z2 x Z ? Justify your answer. 12. Show that G = {1, 7, 17, 23, 49, 55, 65, 71} is a group under multiplication modulo 96. Then express G as an external and an internal direct product of cyclic groups.
Abstract Alg I 1. Can you explain why Z/8Z and the dihedral group D_4 are not...
Abstract Alg I
1. Can you explain why Z/8Z and the dihedral group D_4 are not isomorphic? 2. Consider the subgroup of S_4 generated by the two permutations (12)(34) and (13)(24). Also consider the subgroup generated by (12) and (34). Are these groups isomorphic? Why or why not? Hint: check out the multiplication table
(h) Show that the affine group AGL(1,q) is isomorphic to a subgroup of GL(2,9), the general linear group of non-singular matrices over GF(q), by using the mapping ax + b (Why is this an isomorphism?) [10] (8 h
(2) Consider the following groups: Z24; Z3 x Z7 x Z2; Z2 x Z2 x S3; Zg x Z3; G-symmetries of the squareZ. Which of these groups are isomorphic to one another?
(2) Consider the following groups: Z24; Z3 x Z7 x Z2; Z2 x Z2 x S3; Zg x Z3; G-symmetries of the squareZ. Which of these groups are isomorphic to one another?
How many non-isomorphic unital rings are there of order 4?
Question 3: How many non-isomorphic unital rings R4 are there of order 4? Hint: we can assume that the additive group of R4 can be either (74, +) or (Z2 X Z2, +). Thus the elements of R4 are one or the other of these groups, with a multiplication defined in some way. In the former case, 1 can be assumed to be the multiplicative identity. Why can't 2 be...
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the argument used in lecture to show that R[z]/(x2 1) is isomorphic to C)
4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the argument used in lecture to show that R[z]/(x2 1) is isomorphic to C)
8. (10 points) Consider the general linear group GL(2,26) = {© a) | where a, b, c, d e Zes and ad – be + 0} (a) Determine the order of the group GL(2, Z5). (b) List a Sylow 5-subgroup of GL(2, Z5).
Problem 3. Consider the general linear group GL2 = (M2,*) of 2 x 2 invertible matrices under matrix multiplication. In Homework Problem 9 of Investigation 6, you showed that Pow G 1-( )z is isomorphic to the group Z. Prove that the group (Pow 1 i
problem 4a in worksheet 2
11. Recall from problem 4a on Algebra Problem Sheet 2 that the general linear group GL2(R) is the set of 2 x 2 matrices ahwhere a, b,c,d are real numbers such that ad be 0 under matrix multiplication, which is defined by (a) Prove that the set H-( [劙 adメ0} is a subgroup of GL2(R). (b) Let A = 1] and B-| 의 히 . Show that ord (A)-3, ord (B) = , and ord...
11. Let G = Z4 Z4, H = {0,0), (2,0), (0,2), (2,2)). Write the Cayley table for G/H. Is G/H isomorphic to Z4 or Z2 x Z ? Justify your answer. 12. Show that G = {1, 7, 17, 23, 49, 55, 65, 71} is a group under multiplication modulo 96. Then express G as an external and an internal direct product of cyclic groups.
Abstract Alg I
1. Can you explain why Z/8Z and the dihedral group D_4 are not isomorphic? 2. Consider the subgroup of S_4 generated by the two permutations (12)(34) and (13)(24). Also consider the subgroup generated by (12) and (34). Are these groups isomorphic? Why or why not? Hint: check out the multiplication table
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