Let G = Z[i] = {a+bi | a, b € Z} be the Gaussian integers, which form a group under addition. Let y e R, and define a function p: G + R by p(a+bi) = a+yb. Prove that p is a group homomorphism. Furthermore, show that o is injective if and only if y is irrational.

Q: Consider the group G = {x € R such that x + 0} under the binary operation x*y = -2xy The inverse…

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Q: Let G be a group, let H < G, and let x E G. We use the notation xHx1 to denote the set of elements…

A: In a group G, two element g and h are called conjugate when h = x g x-1 for some x ∈ G For an…

Q: Let G be a group. Using only the definition of a group, prove that for each a E G, its inverse is…

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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 G = {x ∈ R : x 6= −1} . Define △ on G by x△y = x + y + xy   Prove that (G, △) is an abelian…

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Q: Let G be an Abelian group and H = {x E G | |x| is odd}. Prove thatH is a subgroup of G.

A: Given: To prove H is a subgroup of G.

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

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Q: let G be an abelian Group and let H and K be a susb group of G prove that HK ={hk / h an element H…

A: Consider the provided question, Your question is missing the last statement; I think you want to…

Q: Let A = {fm,b : R → R | m ± 0 and fm,b(x) : = mx + b, m, b e R} be the group of affine m b functions…

A: We are given two groups.  A={fm,b:ℝ→ℝ|m≠0 and fm,b(x)=mx+b,m,b∈ℝ} B=mb01 | m,b∈ℝ,m≠0. We need to…

Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D

A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn

Q: Let G be a group. Show that for all a, b E G, (ab)2 = a2b2 G is abelian

A: To prove that the group G is commutative (abelian) under the given conditions

Q: G be defined by f(r) = x1. Prove that f is operation-preserving if 6*. Let G be a group and f: G and…

A: To prove that the given function f is a homomorphism (operation preserving) if and only if G is…

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

A: Multiplication of the elements of the group elements with respect to binary operation

Q: let x be an element of group g. Prove that if |x|=n then x^-1=x^n-1

A: Given 'x' be an element of a group G and |x|=n. As G be a group , inverse of each element of G must…

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: Let G be a group and a E G. Define C(a) = {x € G|ax = xa, for all a E G}. Prove that C(a) < G.

A: A nonempty subset H of a group G is said to be a subgroup of G, if it satisfies the following…

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: LetS=R{−1} and define a binary operationon S  by a∗b=a+b+ab. Prove that (S, ∗) is an abelian group.

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Q: Let G = {x E R |x>0 and x 1}, and define * on G by a * b= a lnb for all a, b E G Prove that the…

A: Detailed explanation mentioned 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: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = –S, |S| = 3 and (S) =…

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

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Q: Consider the group G={x € R such that x#0} under the binary operation *: Th identity element of G is…

A: Solution: Since for any x,y∈G, the operation * is defined as x*y=-2xy The identity element is e=-12…

Q: Consider the group G= (x ER such that x ± 0} under the binary operation * x*y=-2y The inverse…

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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: Consider the group G = {x € R]x # 1} under the binary operation : *• y = xy – x-y +2 The identity…

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Q: Let G be an abelian group,fo f fixed positive integer n, let Gn={a£G/a=x^n for some x£G}.prove that…

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Q: Consider the group G (x ER such that x + 0} under the binary operation **y=-2xy Oxxx-4x^ XX*x-2x^3 O…

A: Using binary operations find the x*x*x

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 be a finite group. Let xeG, and let i>0. Then prove that o(x) gcd(i,0(x))

A: To prove the required identity on the order of the group element

Q: If G is a group with identity e and a2 = e for all a ∈ G, then prove that G is abelian.

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Q: Let p : G → G' be a group homomorphism. (a) If H < G, prove that 4(H) is a subgroup of G' (b) If H <…

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Q: Let G = {2'3m5n : l, m, n E Z}. G is a group of rational numbers under the usual multiplication.…

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Q: Let G be a group with identity element e, and let H and K be subgroups of G. Assume that (i) H and K…

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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 G be a group. Prove that (ab)1= a"b-1 for all a and b in G if and only if G is abelian

A: First, consider that the group is abelian. So here first compute (ab)-1 for a and b belongs to G,…

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: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

A: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

Q: Let G be a group and let x EG of order 23. Prove that there exist an element z EG such that z6 – x.

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Q: 25. Prove that R* x R is a group under the operation defined by (a, b) * (c, d) = (ac, be + d).

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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.

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Q: 1+2n 1- Prove that if (Q – {0},') is a group, and H = { a n, m e Z} 1+2m is a subset of Q – {0},…

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

Q: Let G be an infinite cyclic group. Prove that G (Z,+)

A: To show that any infinite cyclic group is isomorphic to the additive group of integers

Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

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: 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 G be a group and D = {(x, x) | x E G}. Prove D is a subgroup of G.

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Q: Let G be a group. V a, b, c d and x in G, if axb = cxd then ab = cd then G is necessarily:…

A: The answer is given as follows :

Q: Prove that the symmetric group (S₂, 0) is abelian.

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