5. List the distinct elements in (Z/8Z) and (Z/11Z) each of these groups. Write out the...
6. Compute the order of each element in (Z/8Z)* and in (Z/11Z)*.
Consider the following groups of invertible elements: For each group, list its elements. What is the order? Is it cyclic? 「f not, is it isomorphic to some other group you can describe explicitly, e.g. a product Z/nZ x Z/mZ? Consider the following groups of invertible elements: For each group, list its elements. What is the order? Is it cyclic? 「f not, is it isomorphic to some other group you can describe explicitly, e.g. a product Z/nZ x Z/mZ?
0 1 1. Write down the Cayley table for the group generated by the matrices 1 0 -1 0 and 0 1 2. Write down the Cayley table for the permutation group generated by the permutations (12)(34) and (13) in S_4. 3. What do you notice about the two Cayley tables? How do they compare with the Cayley table for Z/8Z? How about the Cayley table for the square? 0 1 1. Write down the Cayley table for the group...
Abstract Algebra I 1. Write down the Cayley table for the group generated -1 0 0 1 by the matrices 1 and 1. 2. Write down the Cayley table for the permutation group generated by the permutations (12)(34) and (13) in S 4 3. What do you notice about the two Cayley tables? How do they compare with the Cayley table for Z/8Z? How about the Cayley table for the square? 1. Write down the Cayley table for the group...
(2) Consider the following groups of invertible elements For each group, list its elements. What is the order? Is it cyclic? If not, is it isomorphic to some other group you can describe explicitly, e.g. a product Z/nZ x Z/mZ?
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
Find ALL the complex solutions to z^10-8z^5+64=0.
Problem 4. (i) Let R> 2/14Z and consider the polynomial ring R[d]. Let A(z) 4 + 2r3 + 3r2 + 4x + 5 and B(x) 37 be elements of R]. Find q(x) and r(x) in R] such that: A(x)-q(z)E(z) + r(z) and deg(r) < 2. (2pts) (ii) Let R- Z/11Z, write down the table of squares in R as follows. For every a E R (there are 11 such elements), find a2. Here you are required to express the final...
Problem 4. (i) Let R> 2/14Z and consider the polynomial ring R[d]. Let A(z) 4 + 2r3 + 3r2 + 4x + 5 and B(x) 37 be elements of R]. Find q(x) and r(x) in R] such that: A(x)-q(z)E(z) + r(z) and deg(r) < 2. (2pts) (ii) Let R- Z/11Z, write down the table of squares in R as follows. For every a E R (there are 11 such elements), find a2. Here you are required to express the final...
problems are related to symmetric groups. 1. In this problem we will give an explicit Cayley embedding of the quaterion group Qs into the symmetric group Sg. First, we fix the following order on the 8 elements in the group: {1,-1,1, -i, j, - j, k, -k}. Now, for each g EQs, write down the corresponding permutation 0, in cycle notation), where o,(h) = gh. Example, 0; = (1 3 2 4)(5 76 8). Now write down o, for the...