Let f: Z→Z be a group homomorphism such that f is one-to-one, then which of the following statements is true: Select one: O a. |Ker(f)| =0. O b.Z/ Im(f)=Z. O cf is also onto. O d. All the above.

Q: Suppose thatf:G G such that f(x) and only if - axa. Then f is a group homomorphism if O a^2 = e a =…

A: See solution below

Q: Prove that the function f: C R++ defined by f(a = bi) = a? + b2 is uniform in a group.

A: Given function f:C-→ℝ++ defined by f(a=bi) = a2+b2. we need to prove f is uniform in a group.

Q: Define on S = Q × Q* the operation (a, b) * (x, y) = (ay+ x, by). Recall that Q* = Q\{0}. 1. Explain…

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Q: Explicitly construct the Galois group for r- 4r² + 2 over Q. To which group is this isomorphic?

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Q: 3. Suppose that ged(m, n) = 1. Define f : Zn Z x Z, by f(r]mn) = ([T]m; [7]n). %3D (a) Prove that f…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

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

A: Given information n=10.

Q: 12. Prove that the following groups are not cyclic: (a) Z2 x Z2 (b) Z2 x Z (c) Z x Z.

A: To show the following groups are not cyclic. If a group is cyclic the there exit an element in that…

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: integer 16. If EG→His a surjective homomorphism of groups and G is abelian, prove that H is abelian.…

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Q: Define on R the operation * by x*y = X+y+k, for all x,y element of R and k is fixed real number. The…

A: We have to check

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…

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Q: be the operation on Z defined by a*b = a+b for all a,beZ. Justify the following questions. 4 Let (1)…

A: Here * be the operation on ℤ defined by a*b=a+b4 for all a, b∈ℤ. We have to justify the followings:…

Q: Let G be a group and let r, y e G such that ya = r-ly. Use the Principle of Mathematical Induction…

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Q: Define the mapping a: R² →R by 7((x,y))=x. (Note that IR is a group under addition with identity 0).…

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Q: Consider the group G = {x € R such that x* 0} under the binary operation x*y=-

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Q: Suppose thatf:G G such that f(x) = axa. Then f is a group homomorphism if and only if O a^2 = e a^4…

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Q: Show that each of the following is not a group. 1. * defined on Z by a*b = |a+b|

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Q: Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

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Q: For the group homomorphism : Zs → Z, defined by ø([r]) = [æ]² for all [r] € Z, find the kernel and…

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

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Q: Let f. → (R, +> be defined by f(x) = 3x- 3 Prove or disprove that f is an isomorphism from the…

A: This is related to isomorphism

Q: consider the Set. H=E34+5m nime Z 27-1Is. idi group of Z Su b

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Q: In the set of real numbers R there is an operation defined as: x x y = Vx³+y³ prove that (R, x) is…

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Q: Show that R* is isomorphic to G?  R* is a group under multiplication G is a group under addition…

A: To show A is one-one Let Ax1=Ax2 where x1 and x2 are two points of R*⇒x1-1=x2-1⇒x1=x2Thus the…

Q: Let f be the homomorphism f Z1s → GL(2) of groups such that (1) = A, where A = a) Give the…

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Q: Consider the group G (x E R]x 1} under the binary operation : ** y = xy-x-y +2 If x E G, then x =…

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

A: Need to evaluate all the abelian group of order 36.

Q: a) Prove that the mapping from U(16) to itself given by x→x Is an Automorphism b) Find the group SG,…

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Q: Prove that each of the following sets, with the indicated operation, is an abelian group. (a) (R, *)…

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Q: Suppose that f: G → G such that f(x) = axa. Then f is a group homomorphism if and only if a = e O…

A: From the condition of group homomorphism we can solve this.

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: Suppose that f:G-G such that f(x)- axa. Then fis a group homomorphism if and only if O a*2e O an4e O…

A: Here we will evaluate the required condition.

Q: 5. Consider the group (R+, 0). Prove that the function F: R -R given by: F (x.y) = (x +y.r-y) is a…

A: F: R2 → R2F (x, y) = (x +y, x-y)Let (x1, y1) ,  (x2, y2) = R2 (x1, y1) +  (x2, y2) =  (x1+ x2, y1+…

Q: Q4: Consider the two group (Z, +) and (R- {0}, ), defined as follow if n EZ, f(n) ={1 if nE Z, %3D…

A: Homomorphism proof : Note Ze denotes even integers and Zo denotes odd integers.  So f(n) = 1 if n is…

Q: 6. Prove the following groups are not cyclic : (a) Z x Z (b) Z6 × Z (c) (Q+, ·) (Here, Q+ = {q € Q…

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

A: We use the following result here. Result: Let n=pq for some prime numbers p and q with p<q. If p…

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

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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 G = {x ∈ R : x != −1} . Define △ on G by x△y = x + y + xy.  Prove that (G, △) is an abelian…

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Q: Let S = {x €R | x + 3}. Define * on S by a * b = 12 - 3a - 3b + ab Prove that (S, *) is a group.

A: The set G with binary operation * is said to form a group if it satisfies the following properties.…

Q: Suppose that f:G - G such that f(x) = axa". Then fis a group homomorphism if and only if O a^3 = e a…

A: f:G→G such that  fx=axa2 We know that f is a homomorphism if fxy=fxfy for all x, y∈G

Q: Let F:Z Z, be a mapping from the group of integers Z to a group of integers mod 6, Z, defined by:…

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

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Q: In the set of real numbers IR there is an operation defined as: I x y= Vr³+y³ %3D prove that (IR, ×)…

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Q: Let F:Z → Z, be a mapping from the group of integers Z to a group of integers mod 6, Z, defined by:…

A: Ker(F) contains elements of Z whose image is zero under mapping F.

Q: Prove that the subgroup {o ES5o (5) = 5} of S5 is isomorphic to $4.

A: The given question is related with abstract algebra. We have to the subgroup σ ∈ S5 | σ5 = 5 of S5…

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: Consider the group G = {x € R such that x # 0} under the binary operation *. ху X * y = x * 2 The…

A: First we have to find the identity element. Let G be the group and e be the identity element of G.…

Q: Show if all primitive transformations of the nonzero form x '= x ,y' = cx + dy d are a group.

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.