Let y: 72 → 14Z be the mapping given by the rule o (a) = 2x for any a e 7Z. a. Show that o is a group homomorphism from the group (72; +) to the group (142; +). b. Is p an isomorphism? Justify your claim. Preview My Answers Submit Answers

Q: 1. If (G,*) a group such that a² = e for all a € G. Prove that G is abelian. 2. Define an operation…

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Q: 2. Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group. (a) Give its…

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Q: 9. Describe the group of the polynomial (x* – 1) e Q[x] over Q.

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Q: 33. If a, bɛR with a # 0, let T:R→ R be the function given by T(x) = ax + b. Prove that the set G =…

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Q: Consider the square X = [-1,1]2 = {(x, y)|x > -1, y < 1} and 0 = (0,0). Show that the fundamental…

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

A: Here we use the definitions of group homomorphism and the kernel of it . Which are given in solution…

Q: Prove if it is a group or not. 1. G = {x € R | 0 < x < 1},x * y = xy 1-x-y+2xy

A: *By Bartleby policy I have to solve only first one as these are all unrelated and very lengthy…

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: Ql: Prove that (Q\{0},x) is a subgroup of (R\{0},x).

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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) =…

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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: ii) Show that the function f (x) defined from the group (R, +) to the (R,×) by f (x) = e* is a…

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

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Q: Q5. Let A and B be two groups. Let 0: A x B → B defined by 0(a, b) = b Is 0 isomorphism? Find…

A: To check whether a function θ is isomorphism, it is required to check  θ is homorphism θ is one-one…

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

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Q: Let G be a group. For a ∈ G define Ta : G −→ G such that Ta(g) = ag for g ∈ G; in words Ta is just…

A: Solution: We know that the permutaion τ on a non empty set A is a bijective map from A to A. That…

Q: I need help with attached abstract algebra question to understand it.

A: To show that the subset H of G is indeed a subgroup of G

Q: 2. Let (G. .) be a group such that a.a = e for all a EG. Show that G is an abelian group.

A: Definition of abelian group : Suppose &lt;G, .&gt; is a group then G is an abelian if and only if…

Q: Let C be a normal subgroup of the group A and let D be a normal subgroup of the group B. Prove that…

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Q: 2. Let G = (1, 0). Decide if G is a group with respect to the operation * defined as follows: x * Y…

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Q: 1. Let 0(V) be the set of all orthogonal transformations on V. Prov O) is a group with respect to…

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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: Let Dn be the dihedral group of order 2n. Decide whether the following pairs of groups are…

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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: . Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group. (a) Give its…

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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: Verify that the nonempty set G with the given operation * is a group. Is G abelian (prove or…

A: Group: A nonempty set G is said to be a group if it satisfies the following properties: Closure:…

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: Q3: (A) Prove that 1. There is no simple group of order 200. 2. Every group of index 2 is normal.

A: Sol1:- Let G be a group of order 200  i.e O(G) = 200                = 5² × 8. G contains k Sylows…

Q: Consi der a normal Subgroup Hof order 4 of the dihedral group Dy= (a, b :a* = b²: (ab)* = 1). Then…

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Q: Let C be a normal subgroup of the group A and let D be a normal subgroup of the group B. Prove that…

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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: 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: How do we describe all the elements in the cyclic subgroup of GL(2, R) generated by the matrix 1 ? 1…

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Q: Show that A4 is nolmal by finding a homomorphism whose kernel is A4.

A: If (alternating group) An is the normal subgroup of Sn, then by first isomorphism theorem implies…

Q: Prove if it is a group or not. 1. G = {x ≤R | 0 < x < 1},x * y = xy 1-x-y+2xy

A: *By Bartleby policy I have to solve only first one as these are all unrelated and very lengthy…

Q: Let Dn (n ≥ 3) be the dihedral group of order 2n. (i) Show that D10 D5 x Z2 by constructing an…

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Q: Consider a dihedral group Dy= (a,b:a'- *. (ab)`=t). (i) Find all the conjugacy classes of Dy Give…

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Q: Let G = {a + b/2|a, b € Z}. Show that G is a group under ordinary addition.

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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^4 = e %3D…

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Q: 3. Show that Q has no subgroup isomorphic to Z2 × Z2.

A: The objective is to show that ℚ has no subgroup isomorphic to ℤ2×ℤ2

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: 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: 22. Prove that the set = {(₁ ~ ) 1} x) | : x, y ≤ R, x² + y² = 1 = SO(2) = forms an abelian group…

A: Given: 22. SO(2)=x-yyx : x, y∈ℝ, x2+y2=1 To show: The given set is a group with respect to…

Q: 25: Let R? = R × R = {(a, b) : a e R, be R} and T: R? → R² s.t. Ta b)(x, y) = (x + a, y + b) (a,.…

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

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