(d) Define the semidirect product Z/5Z × (Z/5Z)* to have Z/5Z × (Z/5Z)* as its underlying set, but with binary operation (a, b) · (c,d) = (a + bc, bd). . Show that Z/5Z × (Z/5Z)* is a group with this operation and that it is nonabelian. (e) Show that o: G(K/Q) → Z/5Z × (Z/5Z)* is a group isomorphism.

Q: Show that the group of positive rational numbers under multiplicationis not cyclic. Why does this…

A: To Show: A group of positive rational numbers under multiplication is not cyclic and why does this…

Q: Show that the group (R#, +) is isomorphic to the group (R#+, *), where R# is the set of real numbers…

A: R# is the set of real number and R#+ is the set of positive real numbers We have to show that the…

Q: Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

A:

Q: 4. Suppose (G₁, *) and (G₂, 0) are groups, let i and J denote the respective identities of G₁ and…

A: and are group. be a group isomorphism.We have to prove that if  has finite order the  has finite…

Q: (G,+) is a finite group such that (a+b)^2 = a^2 + b^2 for all a,b E G. show that G is abelian

A:

Q: Let G - { (a, b) : a eR, be R\{0} }. Define on G the binary operation: (a, b) (c , d)- (a + bc , bd)…

A:

Q: Consider the group G = Z4 X Z9 where the operation is done componentwise, that is (a, b) * (c,d) =…

A:

Q: Let A and B be rings. Define addition and multiplication on the Cartesian product A x B by…

A:

Q: means a cyclic group but 3+i can't turn into 3-i with a positive integer exponent.

A: Please see the answer in next step.

Q: . Consider the group (Z30, +). (a) Find the orders of the elements 3, 5, 6, 7,8, 18, 19, 24 and 25.…

A: The order of an element is n , when n is the least such that, n.a=0 3.10 = 0 and 19 the least , so…

Q: Show that Z is not isomorphic to Zž by showing that the first group has an element of order 4 but…

A: The given groups: Z5x and Z8x That is, Z5x =Number of elements of a group of Z5 Z8x =Number of…

Q: Which of the following sets with an internal binary operation does not have an inverse for every…

A: The objective of the question is to identify which of the given sets does not have an inverse for…

Q: (3) What is the order of (8,2, 5) in group Z12 × Z6 × Z7. (For this question, I don't need a strict…

A:

Q: 9. Classify the following groups in the sense of the Fundamental Theorem of Finitely Gener- ated…

A:

Q: Express the following as products of cyclic gr a) Z140 ») U (140) =) Z400 H) U (400).

A: To express the given groups as products of cyclic groups, we need to find the prime factorization of…

Q: Let P₂(x) = {ax² + b | a, b e R and a #0}. If p(x) = ax² +b and g(x) = cx² + d are elements of…

A: In the given problem we have to show the the set p2(x) is a group with respect to the binary…

Q: 2 Prove That D₂ = 8 cannot be the internal direct product of two of its proper subgroups.

A:

Q: 2. Let R be the set of 2 x 2 matrices with integer entries and lower left entry equal to zero, i.e.:…

A:

Q: True or false, The nonzero elements of ℚ form a cyclic group ℚ* under field multiplication.

A: Cyclic group under Multiplication

Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

Q: Find a matrix M such that M-¹0 (8) M = (19) for every element g € S3. Conclude that the…

A:

Q: Does (U(14),×14) form a cyclic group? If yes, find all generators.

A:

Q: 11. Prove that the set of all 2 × 2 matrices with entries from R and de- terminant +1 is a group…

A: To show that the set of all 2×2 matrices with entries from R and determinant +1 is a group under…

Q: Show that O(n), the set of all orthogonal n × n matrices is a group under matrix multiplication

A: Given: On is the set of all orthogonal matrices of order n×n. i.e. On=A| ATA=AAT=I. We have to prove…

Q: (6) Let (G, *) be a group and a € G. Prove that the order of a is equal to the order of a-1. Prove…

A:

Q: Explain why the group (Z4, +) is NOT isomorphic to the group (LI, ∘) Group (LI ∘): The symmetries…

A: If two groups are isomorphic then they should have same number of elements

Q: In the group U (16) compute, (a) 5-7, and (b) 3-1

A: M(16) refers to th e multiplication group of non-negative integers (mod 16) that are less than 16…

Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D where a = (1 23 4) and b…

A:

Q: (2) (Z x Z)/(6Z × 14Z)

A:

Q: 3. Let D be an integral domain. (a) Prove that FD is an abelian group under the operation of…

A: Let us prove that field of fractions is an abeliean group under addition.