8. Describe all homomorphisms from Z5 into Z, and compute their respective kernels.

Q: Show that the mapping Te data by pla + b/2)=a- bv2, a,beQ is an automorphism of the body Q(/2). Show…

A:

Q: Exercise 8.17. (a) How many (distinct) isomorphisms are there from Z6 to Z2 Z3? (b) How many…

A:

Q: 5. If gcd(m, n) = d, then there are d homomorphisms from Zm to Zn. Identify the four homomorphisms…

A: Given: If gcd(m, n)=d then there are d homomorphisms from ℤm to ℤn. To find: Four homomorphisms from…

Q: Show that U(8) and U(10) are not isomorphic.

A:

Q: Let phi be a homomorphism from Z/30Z to Z/5Z with phi(6) = 2. Determine the kernel of phi.

A:

Q: 7. Show that 4 is a subgroup of S,

A:

Q: Without looking at inner automorphisms of Dn, determine the numberof such automorphisms

A:

Q: 4) Show that if |G| = pq for some primes p and q (not necessarily distinct) then either G is abelian…

A: I haved proved using possible cases of order of Z(G)

Q: Give an example to show that the mapping a --> ap need not be anautomorphism for arbitrary fields…

A:

Q: Find all the homomorphism of (z,+) onto (z,+)

A:

Q: 2) If (1+x) is an idempotent in Zn; Show that (n-x) is an idempotent

A:

Q: Find all homomorphisms from Zg →Z10 , and state the kernel of each of these homorphisms.

A: We have to find all the homomorphisms from ℤ8 to ℤ10 , and we have to state the kernel of each of…

Q: 27. Determine all homomorphisms from Z, to itself.

A:

Q: 4. Show that the collection C = {[a, b) | a < b, a and b rational} is a basis that generates a…

A: Given:         l = {[a,b]} | a&lt;b , a and b rational}

Q: Find all possible automorphisms of Q(/2).

A: We will use basic knowledge of Galois theory and group theory to solve this question.   Note that an…

Q: Determine all ring homomorphism from Zn to Zn

A: (if n is not prime) No. of ring homomorphism from Zn to Zn = No. of Idempotent elements in Zn.

Q: Determine all homomorphisms from Z4 to Z2 ⨁ Z2.

A:

Q: How many homomorphisms are there from Z20 onto Z10? How manyare there to Z10?

A: To find: Number of homomorphisms from Z20 onto Z10 and the number of homomorphisms to Z10.

Q: Let Φ be an automorphism of C*, the group of nonzero complexnumbers under multiplcation. Determine…

A:

Q: Determine all ring homomorphisms from Z to Z.

A: In algebra, A homomorphism is defined as the similarity between the shape, structure, form, etc…

Q: Prove that the group of positive rational numbers, Q+, under multiplication is not cyclic.

A: Group under addition cyclic or non cyclic

Q: Show that the function ø : ℝ → ℝ defined by ø (x) = x3 is an isomorphism of the binary structure ⟨ ℝ…

A:

Q: determine all ring homomorphism from Z20 to Z30

A:

Q: Consider the subset ℚ(sqrt3) = {a + b : a, b ∈ ℚ} of ℝ. Show that ℚ (sqrt3) is a subgroup of ℝ under…

A:

Q: Find all inclusion between subgroups in Z/48Z

A:

Q: Prove or disprove that union of two sigma algebra is a sigma algebra

A: See the attachment.

Q: What is the kernel of homomorphism of \phi:Z4->Z10 defined by \phi(x)=5x?

A:

Q: Show that Z has infinitely many subgroups isomorphic to Z.

A: We have to show that Z has an infinitely many subgroups isomorphic to Z.

Q: Let M, and M2 be two R-modules and N, and N2 be two submodules of M, and M2 respectively. Also, let…

A: Here we need just one fact that if f:X→Y is a function defined from X to Y and if y∈Y, then f-1(y)∈X…

Q: 4 IF Risa Commutativ ring, Show that the characteristic of REX] is the Same as the characteristicof…

A:

Q: Show that if (aib- 2 for all a,b EG draip (G,)is abelian if and od

A: To show: A group G,∘ is abelian if and only if a∘b2=a2∘b2 for all a,b∈G.

Q: Show that the ring (2ot) is principal idenl ring.

A:

Q: Show that the subring of an artinain ring may not be artinian???

A: According to the given information, it is required to show that subring of an Artinian ring may not…

Q: 8. Prove that Q is not a subcartesian product of infinite cyclic groups.

A:

Q: How many ring homomorphisms Z/35→Z/7  are there?

A: Given that ℤ35→ℤ7

Q: Show that U(8) is isomorphic to U(12).

A:

Q: 4) Let : Z--------->Z20 such that (x)= 16x; Show whether is a ring homomorphism or not

A: Given- map ϕ:ℤ→ℤ20 ; ϕ(x)=16x To show whether ϕis a ring homomorphism or not.

Q: Prove that the set of even permutations in Sn form a subgroup of Sn

A: Let E be the set of even permutations in G (which is presumably a group of permutations). Let p and…

Q: What is a mapping of phi defined by the automorphism of Z6?

A:

Q: 4. Construct a 2-dimensional CW-complex whose fundamental group is Z x Z/2 (and prove it).

A: Please check the detailed sol" in next step

Q: Find two p-groups of order 4 that are not isomorphic.

A: Consider the groups ℤ4  and  ℤ2⊕ℤ2. Clearly, both of the above groups are p-groups of order 4.

Q: 1- Show that : is R- module .

A: Note that, in the above definition of Modules, we used the concept of left R - Module &amp; right R…

Q: Find all possible homomorphisms for Z4 Zg and determine the kernel of each of these homomorphisms.

A:

Q: determine all ring homomorphisms from Q to Q

A: Determine all ring homomorphisms from Q to Q.

Q: prove that the set of even permutations in sn forms a subgroup of sn

A:

Q: What are all of the distinct generators of Z17 with respect to multiplication? What are all the…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Suppose that is an isomorphism from Z, O Z, to Z5 and $(2, 3) = 2. Find the element in Z, Z, that…

A: given, ϕ is an isomorphism from ℤ3⊕ℤ5 to ℤ15  and ϕ2,3=2

Q: 8. Prove that Zp has no nontrivial subgroups if p is prime. [#26, 4.5]

A: Follow the steps.

Q: Let phi be a homomorphism from Z/30Z to Z/5Z with phi(6) = 2. Determine the image of phi.

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.