2. Let A and B be two groups with e and e' as identities, then the set P = {(a, b) | a E A, b € B} with the algebraic operation (a, b)(a', b') = (aa', bb') is a group. Prove that the mapping A: B → P defined by A(b) = (e, b) is an embedding of B in P. Also find ker 2.

Q: Show that the frieze group F6 is isomorphic to Z ⨁ Z2

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Q: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

Q: 6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D

A: Fact 1:In a group G, if x∈G such that xn = e, then O(x)|n (e→ identity element.)i.e. order of x…

Q: Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that…

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Q: Show that the center of a group of order 60 cannot have order 4.

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Q: Define ∗ on ℚ+ by a ∗ b =ab/2 . Show that ⟨ℚ+,∗⟩ is a group.

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Q: 7. If x and g are elements of group G, prove that x=g 'xg. Warning: You may not assume that G is…

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Q: Let S, be the set of all one-one and onto mappings on {1,2,3} and Z, = (0, 1,2, 3, 4}. Then, show…

A: Let G be a group and ∗ be an operation defined in G. Then (G, ∗) is group if it satisfies following…

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: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

A: Given that H is a group and h ∈H  Now,we define a mapping f:Z→H  such that   f(n) = hn for n∈Z For…

Q: Z, x is not a group.

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Q: 1. Assume (X, o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) |x € X,y E Y} and define the…

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

A:

Q: What are the three things we need to show to prove that an ordered pair is a group?

A: We have to give the properties of an ordered pair to prove that it is a group.

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: Show that the groups (Z/4, +4) and (Z/5 – {[0]}, x5) are isomorphic.

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Q: Is the set Z* under addition a group? Explain. Give two reasons why the set of odd integers under…

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

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

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Q: Let(G,*) and (H,#) be a groups if f: G H and g: H G are homomorphism such that gof = IG.f og = IH…

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

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Q: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

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Q: 2. Consider the groups (R, +) and (R x R, +). Define the map : RX (RX R) → Rx R defined by r(x, y) =…

A: Given: The group is, (ℝ, +) and (ℝ×ℝ, +) The map is, ℝ×(ℝ×ℝ)→ℝ×ℝ To find: a) The defined map is an…

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

A: Group homomorphism is nothing but a function defined between two groups. The function must be closed…

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

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

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Q: 52

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Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?

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Q: Assume (X,o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) | æ € X, y E Y} and define the…

A: The given question is related to group theory. Given: X , ∘ and Y , ∙ are groups.  Let X × Y = x,y |…

Q: Exercise 2: Let G be a group and a EG. For any m, neZ, prove that am*a = a"a" and (a" y" = am".

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Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of…

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

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Q: Let x be in a group G. If x' - e and x* - e , prove that x - e and x' = e

A: Let G be a group and x∈G.Given: x2≠e and x6=e , where e is the identity element.To Prove: x4≠e and…

Q: 2. Are the groups Z/2Z x Z/12Z and Z/4Z x Z/6Z isomorphic? Why or why not?

A: Here we have to show that given groups are isomorphic

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

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Q: If R2 is the plane considered as an (additive) abelian group, show that any line L through the L in…

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Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…

A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…

Q: .A group (M,*) is said to be abelian if إختر أحد الخيارات (x+y)=(y+x) .a O (y*x)=(x+y) .b O…

A:

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

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Q: 1,) and (G2,*) be two groups and →G2 be an isomorphism. Then *

A: given that G1,. and G2,*are two groups  and φ:G1→G2  be an isomorphism

Q: / Let x be anan- empty set, show that whe ther (P(X), n) is a a group of not

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

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Q: Show that the translations of Rn form a group.

A: we have to show that   the translations of Rn form a group   We know that

Q: Let X be a non-empty set, Show that whether (P(X),n) is a group or not

A: given  Let x be a non empty set, Show that whether  (p(x),∩) is a group or not we need to show it is…

Q: Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x…

A: Let X,◊ and Y,* are two groups. The Cartesian product of X and Y defined by X×Y=x,y | x∈X and  y∈Y.…