Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.
Q: 1. Let G be a group and H a nonempty subset of G. Then H <G if ab-EH whenever a,bEH
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Q: I need help solving/ understanding attached for abstarct algebra dealing with permutations Thanks
A: To prove the property of conjugation of a 3-cycle in the Symmetric group
Q: Let x be an element of group G. Prove that if abs(x) = n for some positive integer n, then x-1 =…
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Q: Let G be a group. Let x EG be such that O(x) = 4. Then: * O (x^12) = 5 O O(x^15) = 5 O O(x^10) = 5…
A: Thanks for the question :)And your upvote will be really appreciable ;)
Q: Let G be a cyclic group ; G=, then (c*b)^=c4* b4 for all a, c, b EG.
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Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D
A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn
Q: If G = is a distant group and f: G → G is an automorph, show that f(a) is a generator of G, i.e. G =
A: We need to show every element of G is some power of f(a)
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: Show that if G is a finite group of even order, then there is an a EG such that a is not the…
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Q: Let h: G G be a group homomorphism, and gEG is an element of order 35. Then the possible order of…
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Q: Show that each of the following is not a group. 1. * defined on Z by a*b = |a+b|
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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: let x be an element of group g. Prove that if |x|=n then x^-1=x^n-1
A: Given 'x' be an element of a group G and |x|=n. As G be a group , inverse of each element of G must…
Q: Suppose H and K be subgroups of a finite group G with |G : H| = m and |G : K| = n. Prove that…
A: We use here, Tower law of subgroup which states that Let (G,∘) be a group. Let H be a subgroup of G…
Q: Let S, be the symmetric group and let a be an element of S, defined by: 1 2 3 4 5 67 8 9 ) B = (7…
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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: For a group G and a fixed element a ∈ G, define the subset C(a) to be the set of all elements in G…
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Q: W6 Assume that H, k, and k are SubgrouPs of the group G and k, , Ka 4 G. if HA k, = HN k Prove that…
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Q: If G is a finite group, H ≤ G, the order of H divides the order of G: | H | / | G | Prove
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Q: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a = a-1.
A: Let G be a group with respect to * . Let e be the identity element,and a is non identity element.…
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 E = Q(√2, √5). What is the order of the group Gal(E/Q)?What is the order of Gal(Q(√10)/Q)?
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Q: 2. Let G be a group. Show that Z(G) = NEG CG(x).
A: Let G be a group. We know Z(G) denotes the center of the group G, CG(x) denotes the centralizer of x…
Q: Let G be a group, and let H < G. Assume that the number of elements in H is half of the number of…
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Q: Q2) If G = Z24 Group a) Is a G=Z24 cyclic? Why b) Find all subgroups of G = Z24 c) Find U,(24)
A: Given that G=ℤ24. a) Then G is generated by the element 1. That is, 1=1,2,3...,22,23,0=ℤ24.…
Q: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion
A: According to the given conditions:
Q: Suppose that a is an element of a group G. Prove that if there is some integer n, n notequalto 0,…
A: Suppose that a is an element of a group G. Prove that if there is some integer n, such that n≠0 and…
Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…
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Q: Label eachof the following statement/s as either true or false. The symmetric group Sn on n…
A: Given: The symmetric group Sn on n elements has order n. To label: The given statement is true or…
Q: (b) Given two groups (G,) and (H, +). Suppose that is a homomorphism of G onto H. For BH and A:{g…
A: Given that G,·and H,* are two groups.
Q: Let G be a group and let a, be G such that la = n and 6| = m. Suppose (a) n (b) = (ea). Prove that…
A: According to the given information, let G be a group.
Q: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.
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Q: Let G be a group and H ≤G. Let x ∈G be an element of finite order n. Prove that if k is the least…
A: It is given that, H is a subgroup of G and |x|=n. Also k is the least positive integer such that…
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 H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.
A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…
Q: Let G be a group and a e G such that o(a) = n < oo. Show that a = a' if and only if k =l mod n. %3D
A: Let G be a group and a∈G such that Oa=n<∞. Show that ak=al if and only if k≡l mod n. If k=l the…
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 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: let G be a group, a,b E G such that bab^-1 =a^r , for some r E N, where N are the natural ones,…
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Q: Let K and H be subgroups of a finite group G with KCHCG.lf [G:K] = 12 and [H:K] = 3. Then, [G:H] =…
A: Let , K and H be subgroups of finite group G. Also . K ⊆ H ⊆ G Here , G : K = 12 , H : K = 3 We…
Q: Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.
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Q: Let G be a group and suppose that (ab)2 = a²b² for all a and b in G. Prove that G is an abelian…
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Q: Let G be a group with identity e and a € G. (a) Define |G|, the order of G, and |al, the order of a.…
A: (a) The cardinality or the number of elements in a group is called the order of that group. If group…
Q: Let H and K be subgroups of a group G and assume |G : H| < +co. Show that |K Kn H G H\
A: Let G be a group and let H and k be two subgroup of G.Assume (G: H) is finite.
Q: according to exercise 27 of section 3.1 the nonzero elements of zn form a group g with respect to…
A: We know that a finite group is said to be cyclic if there exists an element of group such that order…
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…
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Q: Let G, H, and K be finitely generated abelian groups. Prove or disprove: If G × H ∼= K × G, then H…
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- 24. Let be a group and its center. Prove or disprove that if is in, then and are in.Label each of the following statements as either true or false. If x2=e for at least one x in a group G, then x2=e for all xG.Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
- If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.(See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup that leaves each of the elements 1,2,...,i fixed: Ki=gGg(k)=kfork=1,2,...,i For i=1,2,...,n. Prove that G=Sn if and only if HiHj for all pairs i,j such that ij and in1. A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.
- 6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.Label each of the following statements as either true or false. The Cayley table for a group will always be symmetric with respect to the diagonal from upper left to lower right.For each of the following values of n, find all distinct generators of the group Un described in Exercise 11. a. n=7 b. n=5 c. n=11 d. n=13 e. n=17 f. n=19