2. Let G be a group. Prove or disprove that z= { XE G: xg = gx for all ge G} is a Subgroup of G.

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

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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…

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Q: Let G be a group, let H < G, and let x E G. We use the notation xHx1 to denote the set of elements…

A: In a group G, two element g and h are called conjugate when h = x g x-1 for some x ∈ G For an…

Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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Q: Suppose that o: G→G is a group homomorphism. Show that () p(e) = ¢(e') (ii) For every gE G, ($(g))-1…

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Q: 4. Let G be a group and g e G. Prove that the function f: G G given by f(x) = gx is a bijection.

A: The solution for the asked part , is given as

Q: Prove that H x {1} and {1} x K are normal subgroups of H x K, that these subgroups general H x K,…

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Q: 10. Let G be a group and H a subgroup of G. Let CH) - {gEG|gh = hg. VhEH}, Show that C(H) <G.

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Q: Suppose that p:G→G'is a group homomorphism. Show that () p(e) = ¢(e') (1) For every gEG, (ø(g))-l =…

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Q: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: Let (G, ◊) be a group and x = G. Suppose H is a subgroup of G that contains x. Which of the…

A: Solution of the problem as follows

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: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

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Q: Let G be a group and H a normal subgroup of G. Show that if x.V EG Such that xvEH then X,y xyƐH yx…

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Q: Q2)) prove that the center of a group (G, ) is a subgroup of G and find the cent(H) where H = (0, 3,…

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Q: 10. Let (G, *) be a group, and let H≤ G. Define N(H) = {x € G: x¹ *H* x = H} [Normalizer of H in G].…

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Q: 2) Let G be a group and H be a subgroup of G then : a) x • H = y • H + y•x=' € H. b) x • H = H → x e…

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Q: Suppose that 0:G G is a group homomorphism. Show that () o(e) = 0(e) (i) For every gEG, ($(g))¯1…

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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 G be a group and a E G. Define C(a) = {x € G|ax = xa, for all a E G}. Prove that C(a) < G.

A: A nonempty subset H of a group G is said to be a subgroup of G, if it satisfies the following…

Q: Let G be a group and H, KG normal subgroups of G. Prove HnK≤ G.

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Q: t G be a group and let a be one fixed element of G. Show that Ha = {x € G|xa = ax} a subgroup of G.

A: We have to prove if x and y on Ha then xy^-1 lies in Ha

Q: f H and K are two subgroups of a group G, then show that for any a, b ∈ G, either Ha ∩ Kb = ∅ or Ha…

A: If H and K are two subgroups of a group G, then show that for any a, b ∈ G,either Ha ∩ Kb = ∅ or Ha…

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 let H be a subgroup of G with |G : H| = 2. Prove that H a G, that is, H is a…

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Q: 2. Let G be a group. Show that Z(G) = NEG CG(x).

A: Let G be a group. We know Z(G) denotes the center of the group G, CG(x) denotes the centralizer of x…

Q: 2) Let G be a group and H be a subgroup of G then H x = H• y -y-. xcH. true O false

A: (a) Given that G is a group and H is a subgroup of G H.x=H.yH.x.y-1=H.yy-1H.xy-1=Hxy-1∈H Hence,…

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁(G), then H≤ Z(G). me

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Q: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion

A: According to the given conditions:

Q: 2. Let G be a group. Prove or disprove that Z= {x E G: xg= gx for all g€ G} Isa Subgroup of G.

A: To show Z is a subgroup of G, we need to show that (a) Z is non empty (b) For every a , b∈Z , we…

Q: Let G be an abelian group,fo f fixed positive integer n, let Gn={a£G/a=x^n for some x£G}.prove that…

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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 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 p : G → G' be a group homomorphism. (a) If H < G, prove that 4(H) is a subgroup of G' (b) If H <…

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Q: 5. Let G be a group and n e Z+ be fixed. Show that H = {a" | a € G} is a subgroup of G

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Q: Let G be a group. Prove that Z(G) < G.

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

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Q: If G is a non-abelian group of order 8 with Z(G) {e}, prove that |Z(G)| = 2.

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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 and let x EG of order 23. Prove that there exist an element z EG such that z6 – x.

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Q: Let be a group and Ha normal subgroup of G. Show that if y.VEG such that xyEH then yx EH

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Q: Let x, y be elements in a group G. Prove that x^(−1). y^n. x = (x^(−1).yx)^n for all n ∈ Z.

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Q: 1+2n 1- Prove that if (Q – {0},') is a group, and H = { a n, m e Z} 1+2m is a subset of Q – {0},…

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

Q: In the group (Z, +), find (-1), the cyclic subgroup generated by -1. Let G be an abelian group, and…

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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: If H₁ and H₂ be two subgroups of group (G,*), and if H₂ is normal in (G,*) then H₂H₂ is normal in…

A: When a non-empty subset of a group follows all the group axioms under the same binary operation, the…

Q: 3. Let (G, *) be a group and let H and K be subgroups of G. Prove or disprove each of the following…

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Q: Let H and K be subgroups of a group G and assume |G : H| < +co. Show that |K Kn H G H\

A: Let G be a group and let H and k be two subgroup of G.Assume (G: H) is finite.

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

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Q: 2. Let G be a group. Prove or disprove that 7= {x€ G: xg= gx for all gE G} 5a Subgroup of G.

A: This question is related to Abstract Algebra

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.