1. If (G,*) a group such that a² = e for all a € G. Prove that G is abelian. 2. Define an operation on G=R x R\{0} as follows: (a, b) (c,d) = (a + bc, bd) for all (a, b), (c,d) € G. Show that (G, *) is a group. Is G abelian?

Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

A: The solution is given as

Q: , Let G be a finite Abelian group of order n. Suppose G = {91,92,.. {91, 92, 9n} and let x be an…

A:

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: 6. Apply Burnside's formula to compute the number of orbits for the cyclic group G = {(1,5) o (2, 4,…

A: Solve

Q: (Z, +) is a group and infinite group

A: Let a binary operation '*' defined on a set G, then it forms a group (G,*) if it holds the following…

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: (G, .) a group such that a.a = e for all a EG.Show that G is an abelian group. Let be

A:

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

A:

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

A:

Q: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy<minox, oy. Let us take the group…

Q: 6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and…

A:

Q: 7. Let (G, *) be a group and let a G. Define C(a)= {bEG: ab ba Z(G) = {bEG: ab-ba for all a € G}…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

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: w, x) * (y, z) = (wy + 1, xz - 1), for all (w, x), (y, z) ∈ ℚ* × ℤ. 1. Prove that * is a binary…

A:

Q: Let X, Y be cyclic groups of order 2, 3 respectively. Find the orders of the elements of XxY.

A: Let X, Y be cyclic groups of order 2,3 respectively then we have to find the   the orders of the…

Q: 6. Use the Fundamental Homomorphism Theorem to prove that the following groups are isomorphic. (a)…

A: (a) Consider the function           f : Z3 × Z3 → Z3  defined by      f(a, b) = a - b   Then, for…

Q: 1. Define x*y over R\ {-1}by x*y = x + y +xy. Prove that this structure forms an abelian group.

A:

Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

A:

Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

Q: (5) Let (G, *) be a group with a² = e for every a € G. Show that G, a*b = b* a (that is, show that…

A:

Q: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…

A:

Q: Let Ø: Z50 → Z,5 be a group homomorphism with Ø(x) = 4x. Ø-1(4) = %3D O None of the choices O (0,…

A: Here we will find out the required value.

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

A: 2) S=R∖-1 binary operation defined by a*b=a+b+ab

Q: Consider the set G= [0, 1), define an operation on G by if r+y<1 r+y -1 if z+u21 ab= Does the…

A:

Q: 6. ! Write down the relation matrix of the abelian group G specified as follows. G - {x, y, z, w |…

A:

Q: 5. Construct two different Cayley tables for a non-abelian group G = {e, a, b, c, d) where every…

A: Since G={e,a,b,c,d}, so order of G is 5. If for any element x, it is its own inverse then x=x^(-1),…

Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

Q: If G is an abelian group, prove that (a * b)" = a" * b" for all integers n.

A: Given: "G" is an abelian group,  We need to prove that a×bn=an×bn for all integers n. Abelian Group…

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. Show that G = {a+bv2 |a,b e Q} is a group under the usual addition.

A: 3. Let G is said to be group under binary operation (.). It should satisfy the following properties.…

Q: Let Ø: Z50 → Z15 be a group homomorphism with Ø(x) = 4x. Ø-1(4) None of the choices O {1, 16, 31,…

A:

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

A:

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 Ø: Z50 → Z15 be a group homomorphism with Ø(x) = 3x. Then, Ker(Ø) = * (0, 5, 10} None of the…

A:

Q: 5. Let G be a group and define z" z * *** *z for n factors c, where z E G and n E Z+. (a) Suppose…

A: Given that G is a group. Also zn is defined as follows. zn=z*z*z*...*z (a) Use mathematical…

Q: Let f.g:(G,.) (G')and h(m) f(m). g(m), Vm € G, them h is group homomorphism if f is homomorphism O f…

A:

Q: Thus option (c) is correct. Let G be a group with G| = 96. Suppose H and K are subgroups of G with…

A: for abelian group -: condition -: 1 closure ,2 associative , 3 -: identity ,  4 -: inverse ,5-:…

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

A:

Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Let G be a group and H ≤ G. (a) Prove for any a € G, aHa-¹ ≤G. (b) Let x E G and Prove C (x) ≤ G.…

A: Vital instruction : According to Bartleby Guideline , i can solve at most first three subparts if…

Q: 1. Show that the group (a, b, c|bª = a, c¬1 = b) is abelian. %3D

A: The objective is to show that the given group is abelian. Given group is: a,b,c|b4=a,c-1=b

Q: Let (G,) be a group. Show that (G,) is abelian iff (x+y)¹=x-y- for all x, y E G.

A: Given:It is given that  is a group.To Prove:We have to show that the group  is abelian iff  for all…

Q: Suppose a and b are generators of a group G, that is, G = (a) = (b). Show that 4 : G → G given by…

A:

Q: Q)Let G be a group such thatx=x- for each xeG. Show that G is Abelian

A: Given : G is a group such that x=x-1 for each x∈G   To prove : G is abelian.

Q: '. Assume that G is a group such that for all x E G, x * x = e. Prove that G is an abelian group.

A: Consider any two elements a and b in G. So, a,b,ab,ba∈G. Note that I am directly writing the…