b). Let o:Z-Z be given by .0(n)=7n. Prove that o is a group homomorphism. Find the kernel and the image of .o

Q: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: Let m and n be relatively prime positive integers. Prove that the order of the element (1, 1) of the…

A: To prove ℤm×ℤn≅ℤmn ⇔ gcd(m,n)=1 Part 1) Proof of 'only if'  part  ℤm×ℤn≅ℤmn ⇒ℤm×ℤn is cyclic. Let…

Q: Let R = R\ {-1} and define the operation ♡ on R by a♡b = ab + a +b Va, be R. Show that (a) V is a…

A:

Q: Define ∗ on Z by x ∗ y = x + y + 4. Then: (d) Show that every element of Z is invertible with…

A:

Q: Let m, n e Zt such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: We have M and N are abelian groups of order m and n. so, we know that MN is abelian group.

Q: 2- Let (C\{0},.) be the group of non-zero -complex number and let H = { 1,-1, i,-i} prove that (H,.)…

A: To Determine: prove that H,. is a subgroup of a group of non zero complex number under…

Q: Let G be a group with the property that for any x, y, z in the group,xy = zx implies y = z. Prove…

A:

Q: Consider from the b the Symmetric group. 1 2 3 5 6 7 7 5 4 1 2 36 3 5 6 = 3 7 15 a) Express each of…

A:

Q: For each of the following values of n, describe all the abelian groups of order n, up to…

A: For positive integer n, let Cn denote a cyclic group of order n. If G is an abelian group of order…

Q: 3. Let m and n be positive integers such that m divides n. (a) Prove that the map y: Z/nZ Z/mZ…

A: Given m and n are two positive integers such that m divides n. To prove the map φ:ℤ/nℤ→ℤ/mℤ such…

Q: Let U(n) be the group of units in Zn. If n > 2, prove that there is an element k E U(n) such that k2…

A:

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

A:

Q: Suppose n km for positive integers k, m. In the additive group Z/nZ, prove that |[k],| = m, where…

A:

Q: Let U(n) be the group of units in Zn. If n > 2, prove that there is an element k EU (n) such that k2…

A:

Q: and m' be two R-modules. e set Hom, CM, Mm) is an group under pointwise of morphism

A:

Q: Prove that the group G with generators x, y, z and relations z' = z?, x² = x², y* = y? has order 1.

A: In order to solve this question we need to make the set of group G by finding x, y and z.

Q: 5. Let p be a prime. Prove that the group (x, ylx' = yP = (xy)P = 1) is infinite if p > 2, but that…

A: The solution which makes use of matrix theory is presented in detail below.

Q: Consider the group G = {x € R such that x # 0} under the binary operation *: x*y=-2xy O x*x*x=4x^3 O…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

A:

Q: 6. Let H and K be subgroups of G. Suppose |H| = 35 and |K| = 28. Prove %3D that HNK is abelian.

A: The intersection of two subgroups is a subgroup. Also, recall that the prime order subgroup is…

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁(G), then H≤ Z(G). me

A:

Q: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

A:

Q: 3. Use the three Sylow Theorems to prove that no group of order 45 is simple.

A: Simple group: A group G is said to be simple group if it has no proper normal subgroup Note : A…

Q: 3) Show that the subgroup of Dg is isomorphic to V4.

A:

Q: (a) Prove that every element of Q/Z has finite order. (b) Given two groups (G, .) and (H, *).…

A: This question is related to group theory.

Q: Let Dg be the Dihedral group of order 8. Prove that Aut(D8) = D8.

A: We have to solve given problem:

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁1(G), then H≤ Z(G). n Core

A: Given that,  G is group of order 231 and H∈syl11G. We first claim that there is a unique Sylow…

Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

Q: Let S = R\ {−1} and define a binary operation on S by a * b = a+b+ab. (1) Show that a, b ∈ S, a * b…

A: Part A- Given: Let S=R\1 and define binary operation on S by  a*b=a+b+ab To show - a,b∈S,a*b∈S…

Q: Let S= \ {-1} and define an operation on S by a*b = a + b + ab. Prove that (S,*) is an abelian…

A: Given: The operation on S=R\-1 is defined by a*b=a+b+ab To prove: That (S,*) is an abelian group.

Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

Q: Let G = Z[i] = {a+bi | a, b € Z} be the Gaussian integers, which form a group under addition. Let y…

A:

Q: Q:: (A) Prove that 1. There is no simple group of order 200.

A: A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group…

Q: Let S = R\{-1}. Define * on S by a * b = a+b+ ab. Prove that (S, *) is an abelian group.

A:

Q: If I is an ideal of R, then by definition, (I, +) is an abelian group. Consequently, it has an…

A: Since I is ideal , therefore I is definitely a subset of the ring R.

Q: Let ?: ℤ × ℤ → ℝ∗ be defined by p((a, b)) = 2a 3b   (i) Prove that p is a group homomorphism.…

A:

Q: Prove that if B is a subgroup of G then the coset produced by multiplying every element of B with X…

A: Solution: Let us consider (G, .) be a group and B be a subgroup of the group G. Now for any g∈G, the…

Q: Construct the Cayley table for (Zo) ,c), and verify that this is an Abelian group.

A: 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8 0 1 3 4 5 6 7 8 0 1 2 4 5 6 7 8 0 1 2 3…

Q: Prove that is (ab)-1 = a-1b-1 for all a,b in group G, then G is abelian.

A:

Q: 4. Consider the additive group Z. Z Prove that nZ Zn for any neZ+.

A: We know that a group G is said to a cyclic group if there exists an element x of the group G such…

Q: Let x, y be elements in a group G. Prove that x^(−1). y^n. x = (x^(−1).yx)^n for all n ∈ Z.

A:

Q: 7 Suppose that a E G. For each y in the conjugacy class of a, let G(a y)= {g E G: gag= y} be the set…

A: To prove that the given subset G(a -> y) is a coset of the centralizer of a in G.

Q: Let x belong to a group. If x2e while x : x + e and x + e. What can we say about the order of x? =…

A:

Q: For the homomorphism : D4 → ({1, −1}, •) defined by Þ(a) = { 1 -1 if a is a rotation if a is a…

A:

Q: Consider the group G = {x € R such that x # 0} under the binary operation x*y=-2xy O x*x*x=-x^3/4 O…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 22. Prove that the set = {(₁ ~ ) 1} x) | : x, y ≤ R, x² + y² = 1 = SO(2) = forms an abelian group…

A: Given: 22. SO(2)=x-yyx : x, y∈ℝ, x2+y2=1 To show: The given set is a group with respect to…

Q: Let G be a group and suppose that (ab)2 = a²b² for all a and b in G. Prove that G is an abelian…

A:

Q: 1. Consider the groups (R+, ) and (R,+). Then R* and R are isomorphic under the mapping $(x) = log10…

A: We use the definition of cosets, isomorphisms to answer these questions. The detailed answer well…

Q: On Z is defined the following binary operation; x . y = x + y - 1 Show that (Z, . ) is an abelian…

A:

Q: Let R be he set of real mumbers and let bER %3D 1 Show that G is abelian group an under…

A: