Exercise: Let G be a group and {H;, i E 1} is family of subgroups of a group G. Show that Niej H is a subgroup of G. Theorem: Let H.and K two subgroups of a group G. Then HUK is a subgroup of G if and only if either H CK or KCH. Proof: Suppose that either H SCK or K CH.

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

A: Given H is a subgroup of G. We need to find a necessary and sufficient condition for a belongs to…

Q: Exercise 8.3. (a) If H₁ and H₂ are subgroups of groups G₁ and G₂, respectively, prove that H₁ÐH₂ ≤…

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Q: Let Phi be an isomorphism from a group G onto a group H. Prove that phi (Z(G)) phi Z(H) , (i.e. the…

A: Given that phi is an isomorphism from a group G to a group H.Z(G) denote the center of the group G…

Q: Let G, (Isisn) be n groups and G is the external direct product of these groups. Let e, be the…

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Q: c) Show that if G is a group of order 100, then G has at most one subgroup of order 25.

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Q: Let ø : G → G' be an isomorphism of a group (G, *) with a group (G', *'). Prove the following…

A: (A)

Q: a) List all the subgroups of Z, e Zz. b) Is the groups Z, ® Zz and Z, isomorphic? (why?)

A: We use the fact that for distinct prime p and q Zp x Zq is isomorphic to Zpq.

Q: Let G be a cyclic group and H be a subgroup of G. Prove that H is a normal subgroup in G and G/H is…

A: Given:- H be a normal subgroup of a group  G⇒gHg-1=H∀g∈G Also given GH  is a cyclic and we need to…

Q: Let H be a subgroup of a group G and a, b E G. Then be aH if and only if *

A: So,  a, b  belongs to H, and we have  b∈aH  Hence,  b = ah     -- for some element of H  Hence,  a-1…

Q: Let G be a group, H,K ≤ G such that H=, K=for some a,b∈G. That is H and K are cyclic subgroups of G.…

A: Given that G is a group and H, K are subgroups of G with the condition that H=<a> and…

Q: Remark: If (H, ) and (K,) are subgroup of a group (G, ) there fore (HUK, ) need not be a subgroup of…

A: Definition of subgroup: Let (G ,*) be a group and H be a subset of G then H is said to be subgroup…

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 be a subgroup of a group G. Prove that the set HZG) = {hz | h E H, z E Z(G)} is a subgroup…

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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: G let then. [b, a]= be an group and Ta %3D

A: Given that G is a group and also a,b,c∈G. To prove that b,a= a,b-1 Since G is a group, it satisfies…

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 G, and G, be two groups. Let H and H, be normal subgroups of G G, respectively then @ H, x H, 4G…

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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: 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 be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

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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

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Q: {hk | h ∈ H, k ∈ K}}

A: We have to prove that  {hk|h∈H, k∈K} is a subgroup of G.

Q: If G is a group and g E G, the centralizer of g E G, is the set CG(g) := {a E G : ag =ga} that is,…

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Q: Q\ Let (G,+) be a group such that G={(a,b): a,b ER}. Is ({(0,a): aER} ,+) sub group of (G,+).

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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: Give an example of subgroups H and K of a group G such that HKis not a subgroup of G.

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Q: Theorem(7.9): If (H, *) is a subgroup of the group (G, *). then Va e G the pair (a+H a,+) is a…

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Q: Let G be a group which is an algebraic structure.Let H and K be the subgroups of G with H⊂K⊂G. Let…

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Q: (a) Show that given a finite group G and g ∈ G, the subgroup generated by g is itself a group. (b)…

A: PART A: Since, Every subgroup is a group in itself. To prove &lt;g&gt; is a subgroup ,One step…

Q: 4. Let H be a subgroup of a group G. Show that exactly one left coset of H is a subgroup.

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

A: Definition of Homomorphism: Let (G,∘)  and (G',*)be two groups. A mapping ϕ:G→G'is said to be…

Q: Let G Są and let K = {1,(1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. K is a normal subgroup of G. What is…

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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: Let H be a subgroup of a group G and a, be G. Then b E aH if and only if ab-1 e H O ab e H O None of…

A: Ans is given below

Q: a group and H, K be Subgroups of NG (H) = NGCH) Relate H and K? let G be G Such that %3D

A: Given: Let G be the group and H, K be the subgroups of G such that NG(H)=NG(K)

Q: Let G be a group having two finite subgroups H and K such that gcd(|H.K) 1. Show that HOK={e}.…

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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: 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: Although (H,*) and (K,*) are subgroup of a group (G,) then (H * K,*) may field to be subgroup of (G,…

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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: 5. If H. aEA are a family of subgroups of the group G, show that is a subgroup of G.

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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: プーー Let G be a group. H,ksG Sot H= for same a,beG. That is k are cyclic subgroup.of H cund G.Does.…

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Q: Prove that ifH and K are subgroups of a group G with operation *, Question 8. then HNK is a subgroup…

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Q: 1. Let G be a group and let H, H, .. H, be the subgroups of G. The ...

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Q: If a group G is isomorphic to H, prove that Aut(G) is isomorphic toAut(H)

A: We have to prove, If a group is isomorphic to H, then Aut(G) is isomorphic to Aut(H).