Let G be a group, and let X be a set. Let I be the intersection of all subgroups of G that contain X. Show that I is the smallest subgroup of G that contains X. Conclude that 1= (x) I

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: . Let H be a subgroup of R*, the group of nonzero real numbers un- der multiplication. If R* C H C…

A: H be a subgroup of R*, the group of nonzero real numbers under multiplication. R+⊆ H ⊆ R*. To prove:…

Q: Show that if H and K are subgroups of G then so is H ∩ K.

A: Given that H and K are subgroup of group G. We have to show that H∩K is a subgroup of group G.…

Q: Let G be a group and let H and K be subgroups of G so that H is not contained in K and K is not…

A: Given that G be a group and H and K are two subgroup s.t H is not contained in K and  K is not…

Q: Let M be a subgroup of group G, and a,b e G, then aM=bM→ a-1 b € M True O False O

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Q: 15. Let (G, *) be a group, and let H₁, H₂,..., Hk be normal subgroups of G such that H₁ H₂0... Hk =…

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Q: Let G be a group and H, K are subgroups of G with HK=KH. Prove that HK is a subgroup of G.

A: Given that, G be a group and H, K are sub groups of G with HK=KH. Let x∈HK. Then x=hk for some…

Q: Let H be a subgroup of a group G and a, be G. Then bE aH if and only if * O a-1b eH O ab-1 eH O None…

A: We know that  b∈bH                    (1) We know that  aH = bH if and only if a-1b ∈H…

Q: Let G be a and group Z(G)=< g€G•xg=gx, VX€G} %3D be the center of G. Show that G is commutative if…

A: Solve the following

Q: (a) Prove that if K is a subgroup of G and L is a subgroup of H, then K x L is a subgroup of G x H.

A: The detailed solution of (a) is as follows below:

Q: Let H and K be subgroups of a finite group G with H C KC G. Prove that |G:HI |G:K| |K:H].

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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: that a and xax^-1 have same order for all x belongs G.

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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: If H is a Sylow p-subgroup of a group, prove that N(N(H)) = N(H).

A: Let G be a finite group and H be the subset of G. Then, normalizer of H in G, when we conjugate H…

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 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 be a group, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

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: H be a subgroup of G.

A: We have to find out the truth value of the given statements. It is given that H is a subgroup of G.…

Q: Let G be a group with |G|=187 then every proper subgroup of G is:

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Q: Let G be a group and let H and K be subgroups of G. Prove that the intersection of H and K, H n K =…

A: 1. It is given that H∩K=x∈G|x∈H and x∈K Let  x, y∈H∩K ⇒x,y∈H and x,y∈K⇒xy-1∈H and xy-1∈K⇒xy-1∈H∩K…

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 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 be the additive group Rx R and H = {(x,x) : x E R} be a subgroup of G. Give a geometric…

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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 H and K be normal subgroups of a group G such nat HCK, show that K/H is a normal subgroup of…

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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 and K are subgroups of a group G, prove that ANB is a subgroup of G.

A: GIVEN if H and K are the subgroup of a G, prove that  A∩B  is a subgroup of G

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: Show that if aH=H then a belongs to H. H is a subgroup of a group G and a is an element of G

A:

Q: Let H be a subgroup of a group G and a, bEG. Then bE aH if and only if * O None of these O ab e H O…

A: here option (c) is true.

Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

Q: Let H and K be two subgroups of a group G. Let HK={ab|a∈H,b∈K}. Then HK is a subgroup of G. true or…

A: F hv

Q: Suppose that G is a finite group and let H and K be subgroups of G. Prove that |HK| = |H||K|/|HN K|.

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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: 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: 7. Let G be a group, and let g E G. Define the centralizer, Z(g), of g in G to be the subset Z(g) =…

A: let G be a group, and let g∈G. Define the centralizer, Zg of g in G to bethe subset…

Q: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

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Q: Let H and K be subgroups of a group G and assume |G : H| = +0. Show that |K Kn H |G HI if and only…

A: Given:

Q: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

A: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

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 (G,*) be any group and (a) = {a'| i = 0, +1, F2, F: (a) = {... , a-2, a-1, a° = e, %3D %3D…

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Q: 2) Let (G, *) be a group and H, K be subgroups in G. Prove that subset H * K is a subgroup if and…

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Q: Let H be a subgroup of G. If G has exactly one subgroup of order |H|, then show that for all g e G,…

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Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

Q: Let G be a group and D = {(x, x) | x E G}. Prove D is a subgroup of G.

A:

Q: Let H and K be subgroups of a finite group G. Show that |HK |HK= |HОКI where HK (hk hE H, k E K}.…

A: let D = H ∩K then D is a subgroup of k and there exist a decomposition of k into disjoint right…

Q: Let H be a subgroup of G such that x^2 ∈ H for all x ∈ G, then show that H is a normal subgroup of…

A: H = {x² : x ∈ G} And, H &lt; G