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

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

A:

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: E If (H, *) is a subgroup of the group (G, *). then va e G the pair (a' H *a,*) is a subgroup of (G,…

A: Given below the proof

Q: If H is a subgroup of a group G such that (aH)(Hb) for any a, bEG is either a left or a right coset…

A:

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 a be an element of a group G such that Ord(a) = 30. If H is a normal subgroup of G, then Ord(aH)…

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Q: 9) Let H be a subgroup of a group G and a, be G. Then a e bH if and only if O b-la e H O ba e H O…

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Q: For any subset H of a group G, if ab^-1 is in H for all a,b element of H then H is a subgroup of G.…

A:

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

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

Q: Theorem: Let (K,+)is a subgroup of a group (H, ) and (H,) is a subgroup of a group (G,) then (K, )is…

A:

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: 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: Suppose S is a nonempty subset of a group G.(a) Prove that if S is finite and closed under the…

A: (a)Suppose S is a non-empty subset of a group G. then we have to prove that if S is finite and…

Q: Let H be a subgroup of a group G and a, b EG. Then b E aH if and only if O None of these O ab EH О…

A: The solution is :

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: Every subgroup of a group G is normal * False True

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

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A: Given orders of subgroup 10 18 30

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

A:

Q: If psi is homomorphism of group G onto G bar with kernal K and N bar is a normal subgroup of G bar.…

A: Introduction: If there exists a bijective map θ:G→G' for two given groups G and G', then θ is…

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.

A:

Q: The set A := {(1), (1 2 3), (2 3 4)} forms a subgroup of the permutation group (G,.). O True O False

A:

Q: If a simple group G has a subgroup K that is a normal subgroup oftwo distinct maximal subgroups,…

A: Here given G is simple group and K is a normal subgroup of G. Then use the definition of simple…

Q: 9) Let H be a subgroup of a group G and a, be G. Then a e bH if and only if O None of these O b-1a e…

A:

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: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * ba-1 E H ba E H O b-1a…

A: Q9. Third option is correct.

Q: Let H and K be subgroups of a group G. Prove that HNK is a subgroup of G.

A:

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: 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: 1) If (H, *) is a subgroups of (G, *)then (NG(H) , * ) is a subgroup of (G, *).

A:

Q: A subset of H of a group G is a subgroup of G if the operation on G makes H into a group. Prove that…

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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…

A:

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * None of these b-1a e H…

A: Second option is correct.

Q: If a subgroup H of a group G is cyclic, then G must be cyclic. Select one: O True O False

A: we will give the counter example in support of our answer.

Q: Let a be an element of a group G such that Ord(a) = 32. If H is a normal subgroup of G, then Ord(aH)…

A: Result: Let G be a group and H be a normal subgroup of G. Let 'a' be an element of G such that order…

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: (a) If G is abelian and A and B are subgroups of G, prove that AB is a subgroup of G. (b) Give an…

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Q: Let H be a subgroup of G and let a, be G. If Ha Hb, then* %3D aH = bH O a-1H = b-1H O Ha = Hb Ha-1 =…

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Q: Let a be an element of a group G such that Ord(a) = 30. If H is a normal subgroup of G, then Ord(aH)…

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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…

A:

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

A:

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).