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) could be * O 6 O None of the choices O 4 O 8

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: 3. Let G be a group, H a subgroup of G, and g and element of G. (a) Prove that {ghg-1 | h e H} is…

A: Let G be a group, H is a subgroup of G, and g be an element of G

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A: The given set of matrix is 100a10bc1 where a, b, c∈ℤ2. To find: the group to which the given set is…

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: 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: If K is a normal subgroup of a finite group G and S is a Sylow p-suby

A: Given that if K is a normal subgroup of a finite group  G and S is a Sylow p-subgroup of G. then K∩S…

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.…

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Q: Let A be a subset of the group G. Prove that the normalizer of A, NG(A) = {g e G: gAg=A }, is a…

A: Consider the provided question, According to you we have to solve only question no. 2. (2)

Q: Suppose that G is a group and |G| = pnm, where p is prime andp >m. Prove that a Sylow p-subgroup…

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Q: Show that if H and K are subgroups of a group G, then their intersection H ∩ K is also a subgroup of…

A: Subgroup Test           A subset H C G of the group G will be a subgroup if it satisfies the…

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 G is a finite group with |G|<160 and G has subgroups of orders 1O, 16 and 20 then the order of G…

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Q: 4. Let G, Q be groups, ɛ: G → Q a homomorphism. Prove or disprove the following. (a) For every…

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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: Suppose that 0: G G 5a group homomorphism. Show that 0 $(e) = 0(e) (ii) For every geG, (0(g))= 0(g)*…

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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: As you asked multiple questions , I answered only first question.   Here we can say from a corollary…

Q: There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two…

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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: If G is a finite group with |G|<120 and G has subgroups of orders 1O, 15 and 20 then the order of G…

A: Using Lagrange's theorem it can be written that if G is a finite group and B is a subgroup of G,…

Q: If G is a finite group with |Gl<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Given: The group G is a finite group with | G | &lt; 120 and G has subgroups of orders 10, 15, and…

Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

Q: Let n be a positive integer. Show that A, is a normal subgroup of S, by choosing an appropriate…

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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 be a group of order 24. If H is a subgroup of G, what are all the possible orders of H?

A: Given, o(G)=24 wherre H is a subgroup of G from lagrange's theoram: for any finite order group of G…

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

A: H is normal subgroup of G. And a belongs to G.  O( aH) = 3 in G/H and O(aH) in G/H divides  O(a) in…

Q: 5. Let H and K be normal subgroups of a group G such that H nK = {1}. Show that hk = kh for all h e…

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Q: 6. (b) For each normal subgroup H of Dg, find the isomorphism type of its corresponding quotient…

A: First consider the trivial normal subgroup D8. The quotient group D8D8=D8 and hence it is isomorphic…

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…

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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 let G N Subgroup be be of G a a group and normal of finite

A: To prove that H is contained in N, we first prove this: Lemma: Let G be a group.H⊂G. Suppose, x be…

Q: If G is a finite group with IG|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

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Q: If G is a finite group with IG|<160 and G has subgroups of orders 10, 16 and 20 then the order of G…

A: Given :- order of a group G... |G| &lt; 160 Also order of subgroups of group G are 10, 16 , 20. We…

Q: Let H and K be subgroups of the group G, and let a, b E G. Show that either aH n bK = Ø or else aH N…

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Q: Let A be a subset of the group G. Prove that the normalizer of A, NG(A) = { g e G : gAg-1 = A}, is a…

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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: If G is a finite group with IG|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Using Lagrange's theorem it can be written that if G is a finite group and A is a subgroup of G,…

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: If H is a subgroup of G such that [G : H] = 2, then show that H is a normal subgroup of G.

A: Suppose H≤G such that [G:H] = 2. Thus H has two left cosets (and two right cosets) in G.

Q: Let G be a group and g E G. Prove that if H is a Sylow p-group of G, then so is gHg-1

A: It is given that, G is a group and g∈G. To sow that if H is a sylow p-subgroup of G, then so is…

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

A: Use the fact that order of subgroup divides order of group

Q: Suppose that G is a group and |G| = pnm, where p is prime and p &gt; m. Prove that a Sylow…

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Q: If H is the subgroup of group G where G is the additive group of integers and H = {6x | x is the…

A: Let H is a subgroup of order 6 . Take H=6Z where Z is integers.

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

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Q: Let G be a group and let a E Ga G with a = 8. the order of a² is not equal to the order of the…

A: The given statement is

Q: If G is a finite group with |G|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A:

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)…

A:

Q: Suppose that G is a group such that Ord(G) = 36. The number of subgroups %3D that G has is 4 О 12 О…

A: Order of a group: Let G be a group and n be the number of elements in the group. Then, order of…