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 and g are homomorphism g is homomorphism other

Q: let G be an abelian group. And let H = {r :z€ G) show that H < G? %3D

A:

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

A: As per our guidelines we are supposed to answer only one asked question.kindly repost the other…

Q: (a) Let p: G → H be a group homomorphism. Show |p(x)| < |x| for all x E G.

A:

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

A:

Q: Let G be a group such that a^2 = e for each a e G. Then G is * О Сyclic O None of these O…

A:

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

A:

Q: Let f:G-G be a group homomorphism then H = {a € G:f(a) = a} is subgroup O True False

A:

Q: Theorem Let f: G H be a group homomorphism. Then, Im f≤ H.

A: Let us consider the mapping f:G→H . Then f is group homomorphism if  f(x·y)=f(x)·f(y) where, x,y∈G.…

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: Let G = {[1 0 0 1] ,[−1 0 0 1] ,[1 0 0 −1], [−1 0 0 −1]} . Is G ∼= K4? If yes, give an explicit…

A:

Q: Consider the map o : G¡ → G2 defined by: 9(a) = a-! %3D (a) Does o define group homomorphism? (b)…

A: Property of homomorphism of groups....

Q: GX H G, X H. 19. Prove that a group Gis abelian if and only if the function f:G→ G given by f(x) =…

A:

Q: Let h: G G be a group homomorphism, and gEG is an element of order 35. Then the possible order of…

A:

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

A:

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: Suppose thatf: G → G such that f(x) = axa². Then f is a group homomorphism if and only if ) a^2 = e…

A: Option C.

Q: Let G be a group. Define ø : G → G by ø(x) = x-1 for all x E G. (a) Prove that ø is one-to-one and…

A:

Q: 3. Let f : (R\{0},-) (R\{0}, -) be group homomorphism defined by f(a) = |a|. Then ker(f) %3D =...

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: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a^2= e a =…

A: Since f is a group homomorphism , where f(x)=a∗x∗a−1,    x∈G. So a^-1=a implies self inverse implies…

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: Let G be a group and define the map ø : G → G by $(9) = g¬1. Show that o is an automorphism if and…

A: The solution is given as

Q: 3 be group homomorphisms. Prove th. = ker(ø) C ker(ø o $).

A:

Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

A:

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: Let f: - be defined by f(x) = 3x-3. Prove or disprove that f is an isomorphism from the additive…

A: Consider the given information: Let f:ℝ,+→(ℝ,+) be defined by, f(x) =3x-3 To find that f is an…

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

A: We will use property of homomorphism to solve the following question

Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A:

Q: (b) Given two groups (G,) and (H, +). Suppose that is a homomorphism of G onto H. For BH and A:{g…

A: Given that G,·and H,* are two groups.

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

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: Prove that An even permutation is group w.y.t compostin Compostin function.

A:

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

A:

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

A: Given that f from G to G is a function defined by f(x)=axa2 Then we need to find a necessary and…

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: Determine whether an onto homomorphism between the groups D6 and D3 + Z2 exists.

A:

Q: 2* Let f G H be a group homomorphism. Prove that if x E G and n is a natural number then f(x)= f(x)"

A: To prove the required property of group homomorphisms

Q: A group homomorphism f:G G'is called Epimorphisum if f is 1-1 False True

A: We have to tell A group homomorphism f : G → G' is called Epimorphism if f is 1-1  Is true or false…

Q: 9. Show that the two groups (R', +) and (R' – {0}, -) are not isomorphic.

A:

Q: Suppose that f:G G such that f(x) : and only if = axa. Then fis a group homomorphism if a^2 = e

A: A mapping f from a group (A,.)  to a group (B,*) is called a group homomorphism if f preserves the…

Q: G be the external direct product of groups G, G2.. H, = {4,e2.e*, e..e,x, e G,}

A:

Q: 1. AUT(G) := {p: G → G|p is an automorphism} Prove: (AUT(G),º) is a group (under composition)

A: Given: Aut(G)={ φ:G→G| φ is an automorphism}

Q: Let 0:Z50-Z15 be a group homomorphism with 0(x)=4x. Then, Ker(Ø)= {0, 10, 20, 30, 40)

A:

Q: 2. Let H and K be subgroups of the group G. (a) For x, y E G, define x ~ y if x = hyk for some h e H…

A:

Q: Let ø:Z50→Z15 be a group homomorphism with ø(x)=4x. Then, Ker(ø)= * None of the choices

A: