Suppose 0: G-→ G be a homomorphism of groups. Show that: (i) Kere is normal in G (ii) The mappping o: G/ Ker0 →0(G) defined by þ(gKer0) = 0(g) is an isomorphism of groups.

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

A: We have given a map ,  ϕ : A → B , where A = ℝ , +   ,  B = ℝ* , · such that  , ϕx = 2x We know that…

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: Let G be a group of order 24. Show that G is solvable

A: Let G be a group of order 24. Show that G is solvable

Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

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Q: (a) of G'. Show that if y :G → G' is a group homomorphism then Im(y) is a subgroup

A: According to the given information, For part (a) it is required to show that:

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…

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Q: Prove that a group of order 12 must have an element of order 2.

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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: integer 16. If EG→His a surjective homomorphism of groups and G is abelian, prove that H is abelian.…

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Q: is] Let G and H be groups, and let T:G→H_be Isomorphism. Show that if G is abelian then H is also…

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

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: 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: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

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Q: In the following problems, let G be an abelian group and prove that the set H described is a…

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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…

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Q: Suppose that f: G G such that f(x) = axa*. Then f is a group homomorphism if %3D and only if a = e O…

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Q: Let f. → (R, +> be defined by f(x) = 3x- 3 Prove or disprove that f is an isomorphism from the…

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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: Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your…

A: Lets solve the question.

Q: If f: G to H is a surjective homomorphism of groups and G is abelian, prove that H is abelian.

A: As we know that a group homomorphism f:G to H is a map from G to H satisfying:

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

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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 G be a group and let Z(G) be the center of G. Then the factor group G/Z(G) is isomorphic to the…

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

A: Automorphism and abelian group

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Q: Show that the quotient group Q/Z is isomorphic to the direct sum of prufer group

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Q: Show that the multiplicative group Z is isomorphic to the group Z2 X Z2 8,

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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: Suppose that 0: G G is a group homomorphism. Show that 0 $(e) = ¢(e') (1) For every gEG,…

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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…

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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: Let G be a group such that a^2 = e for each aEG. Then G is * Non-abelian Cylic Finite Abelian

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Q: Let G be a non-trivial group. Prove that Aut(G) × Aut(G) is Aut(G x G). a proper subgroup of

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Q: If A is an abelian group with A <G and B is any subgroup of G, prove that ANB < AB.

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Q: Suppose that ¢ : G → G' is a group homomorphism and there is a group homomorphism : G' → G such that…

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Q: Let G and H be groups, and let ø:G-> H be a group homomorphism. For xeG, prove that )x).

A: Given:G and H be the groups.

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: 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…

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Q: Show that every group G of order n is isomorphic to a subgroup of Sn. (This is also called Caley's…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.