Let H be a subgroup of G. If G has exactly one subgroup of order |H|, then show that for all g e G, gH = Hg. Hint: First, show that g Hg- is a subgroup of G where ge G then use this fact.

Q: Recall that the center of a group G is the set {x € G | xg = gx for all g e G}. Prove that he center…

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Q: Suppose that H is a subgroup of G and let G act by conjugation on the set X = {gHg-1 : g ∈ G}.…

A: Given that,H is a subgroup of G and G act by conjunction on the set

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

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Q: be a group and Ha normal subgroup of G. Show that if x,y EG such that xyEH then yxEH Let G

A: Given: Let G be a group and H a normal subgroup of G.To show that x,y∈G suchthat xy∈H then yx∈H

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: If H is a normal subgroup of G and |H| = 2, prove that H is containedin the center of G.

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Q: Let G be a group, let H be a proper subgroup of G, and let a E G. Prove that the left coset aH is a…

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Q: If H is a proper subgroup of L and L is a proper subgroup of G. If |G| = 1250 and |L| = 125, list…

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Q: Prove If S1 and S2 are subgroups of G, then S1 intersection  S2 is a subgroup of G.

A: Let S1 and S2 are two subgroups Then if x, y E S1 or S2 .xy E S1 or S2 And V x E S1 or S2 Then x-1 E…

Q: 53. Let H be a subgroup of G and C(H) = {g €G|gh = hg for all he H}. Prove C(H) is a subgroup of G.…

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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: Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

Q: Let G be a group, and let X be a set. Let I be the intersection of all subgroups of G that contain…

A: Let G be a group and X be a set in G. Suppose I is the intersection of all subgroups of G that…

Q: Let H and K be subgroups of a group G. (a) Define HK = {hk | he H, ke K}. Show that if K is normal…

A: We will solve all the three parts. Given that  H and  K are subgroup of G

Q: Let G be an abelian group and suppose that H and K are subgroups of G. Show that the following set…

A: According to the given information, Let G be an abelian group and suppose that H and K are subgroups…

Q: Let G be a group and a e G. Prove that C(a) is a subgroup of G. Furthermore, prove that Z(G) = NaeG…

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Q: Let G and H be groups. Let p : G → H be a homomorphism and let E < H be a subgroup. Prove that p(E)…

A: Given: φ:G→H is a group homomorphism and E≤H. To prove: a) φ-1(E)≤G b) If E ⊲ H then φ-1E ⊲ G

Q: Let H be a subgroup of G and let a, b E G. If aH = bH, then * На 1 — НЬ 1 О На %3Dнь На-1 %3D НЬ-1…

A: H is a subgroup of G and a,b belongs to G. If aH=bH then a and b lies in the same left coset of H.…

Q: Let G be a group and the center of G is defined as Z(G) = {x E G | xg = gx for all g E G} We already…

A: Let  G be a group and the center of G is defined as  ZG=x∈G|xg=gx for all g∈G⋯⋯(1)  And ZG is a…

Q: Let G and H be groups and let ø:G → H be a homomorphism. Prove that ø(G), the image of ø, is a…

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Q: Let G be a group and H a normal subgroup of G. Show that if x,y EG Such that xyEH then 'yx€H-

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Q: Let Ha normal subgroup of G. Show that if x.v EG Such that xyEHthen yxEH- be a group and Attach File…

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Q: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

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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: Let G and H be groups and let :G →H be a homomorphism. a) Prove that (G), the image of p, is a…

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Q: Let G be a group and let H and K subgroup of G. Prove that the intersection H and K is a subgroup of…

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Q: Let G and H be groups, and let 0 : G -» H be a {x E G 0(x) = e} is called the kernel of 0 and is…

A: From the given information:Let G and H be the groups

Q: Let K be a subgroup of G and let H = {g : gKg^-1} = K. Show that H is a subgroup of G

A: Let K be a subgroup of G and H be defined as H = { g : gKg-1 = K }. Then, we have to show that H is…

Q: If H≤G and let C(H) = {x element G| xh=hx for all h element H} prove that C(H) is a subgroup of G.

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Q: let H be a normal subgroup of G and let a belong to G . if the element aH has order 3 in the group…

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Q: Let G =U(9) and H= (8). Explain why H is a normal subgroup of and construct the group table for the…

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Q: et H < G. Then Gµ = {g € G | gHg¬ = H} is a subgroup of G that has H has a -ormal subgroup. %3D

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Q: Prove that C(a) is a subgroup of G.

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Q: 3) Let H be a subgroup of G then H be a normal subgroup of G if there exist gE G such that g • H •…

A: 3) Let H be a group of G then H be normal subgroup of G if there exist g∈G such that g*H*g-1=H…

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) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: Let H < G. Recall that NG(H) = {g € G: gHg¯l = H}. 1). Prove that H 4 N(H). 2). If K is a subgroup…

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Q: Let φ : G → H be a group homomorphism. (a) Prove that Ker(φ) is a normal subgroup of G. (a) Prove…

A: To discuss normality of kernel and image under group homomorphisms,

Q: Let G be a group, let H≤G be a subgroup, and let N G be a normal subgroup. (i) Show that HnN is a…

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Q: let x be a nonempty G-set and let You be a nonempty subset of x. let Gy={g € G | gy=y for all your…

A: Let X be a non-empty G-set. Also , Y be a non-empty subset of X. We have a set , GY = g∈G / gy = y ,…

Q: Let H be a subgroup of G. If g e G, show that gHg-1 = {ghg-1|he H} is also a subgroup of G. This…

A: We have to show that the set gHg-1 is a subgroup of the group G, where gHg-1=ghg-1  h∈H. We know…

Q: Let H be a subgroup of G and let g be an element of G. the conjugate subgroup gHg^-1 is defined to…

A: We have to prove that gHg^-1 is a subgroup of G

Q: Let G = D8 and H = . Show that H is a normal subgroup of G and list the elements of the quotient…

A: We are given the group G to be Dihedral group D8 and H = &lt;r2&gt;.  First, we need to show that  H…

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: a. If G is a group of order 175, show that GIH=Z, where H is a normal subgroup of G. b. Show that Z…

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Q: Prove that if H is normal in G and ø is onto, then ø[H] is normal in G'.

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Q: Let H be a subgroup of G, define C(H) the centralizer of H.

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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.