If G is a finite group with IG|<120 and G has subgroups of orders 10, 15 and 20 then the order of G is: * O 90 O 30 80
Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…
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Q: Let G be a group and a E G be a certain fixed element of G. The centralizer of a in G is C(a) = {g €…
A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…
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
Q: O If a group G acts on a set S, every element of S is fixed by the identity of G. O Every group of…
A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…
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: Suppose that G is a cyclic group such that Ord(G) = 48. The number of subgroups that G has is * O 8…
A: Q1. Third option is correct. Q2. Second option is correct.
Q: Let G be a finite cyclic group and a be an element of G of order 8 then the order of G would be: O…
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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: 5. Let p and q be two prime numbers, and let G be a group of order pq. Show that every proper…
A: We have to prove that: Every proper subgroup of G is cyclic. Where order of G is pq and p , q are…
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: 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: 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: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * ba EH O None of these…
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Q: Let G be a finite group of order n, positive integer, with identity e. Prove or disprove: For any g…
A: Let G be a finite group of order n, with identity e i.e, order of (G) = |G| = n . Now let g be any…
Q: Let G be a finite group and H1, H2,…., Hk be subgroups of G. .... (a) Show that N H; = Hị n H2 n..n…
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Q: 9) Let H be a subgroup of a group G and a, b E G. Then a E bH if and only if O b-1a E H O None of…
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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 ba e H O b-1a e H…
A: We will use definition of left coset
Q: Let G be a group and a ∈ G. The centralizer of a in G is equal to the centralizer of a^-1 in G.…
A: First let us see the definition of centralizer or normalizer of a in G (definition from I.N.…
Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.
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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: Let G be a group and a e G. Show that o(a) = o(a-). order n, then ba also has order n.
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Q: Let K and H be subgroups of a finite group G with KCHCG. If [G:H] = 4 and [H:K] = 3. Then, [G:K] =…
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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 | < 120 and G has subgroups of orders 10, 15, and…
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 K and H be subgroups of a finite group G with KCHCG.If[G:H] = 4 and [H:K] = 3. Then, [G:K] = 3 4…
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Q: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.
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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 be a group and a be an element of this group such that a^63e. The possible orders of a are: *…
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Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10…
A: If G is cyclic group and order of G is 'n'. Then number of subgroups of G is equal to number of…
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: 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…
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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 * ba-1 E H ba E H O b-1a…
A: Q9. Third option is correct.
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| < 160 Also order of subgroups of group G are 10, 16 , 20. We…
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 finite cyclic group and a be an element of G of order 8 then the order of G would be: 2…
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Q: 189. Let be given Ga finite group and Pe Syl,(G). Give an example of a subgroup H of G where HnP is…
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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: Let G be a finite cyclic group and a be an element of G of order 9 then the order of G would be: * 4…
A: Fourth option is correct.
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: 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: 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: If H and K are subgroups of a group G then H n K is a subgroup of G.
A: Note: according to our guidelines we can answer first question and rest can be reposted. Lemma:…
Q: 5. Let G be the symmetric group S3. Calculate NG(H) when H is i. the subgroup {1, (12)} ii. the…
A: The normalizer NGH of a subgroup H of a group G can be defined to be a set NGH=g∈G gHg-1=H or…
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: 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: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10 O…
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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: Suppose that G is a group such that Ord(G) = 36. The number of subgroups that G has is 4 O 12 O 18…
A: Given order of G is 36 So U(G) = {1,5,7,11,13,17,19,23,25,29,31,35} So number of elements are 12…
Q: 1. Let G be a group and let H, H, .. H, be the subgroups of G. The ...
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- Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.13. Assume that are subgroups of the abelian group . Prove that if and only if is generated by11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.
- Let H1 and H2 be cyclic subgroups of the abelian group G, where H1H2=0. Prove that H1H2 is cyclic if and only if H1 and H2 are relatively prime.5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .
- Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.Prove or disprove that H={ [ 1a01 ]|a } is a normal subgroup of the special linear group SL(2,).let Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n are relatively prime. Exercise16 For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication.