Show that each of the following is not a group. 1. * defined on Z by a*b = |a+b|
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Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.
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Q: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.
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Q: Let G be a group, and let a E G. Prove that C(a) = C(a-1).
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Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…
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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: 1. Assume (X, o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) |x € X,y E Y} and define the…
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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.
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Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…
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Q: Prove or Disprove each of the following. [a-i] The group Z2 × Z3 is cyclic.
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Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, Cg(A) = A and…
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Q: Let G = (Z,, +6) is an Abelian group then how many self - invertible elements in G? (A) 1 (B) 2 (C)…
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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: Let ?1 , ?2 ??? ?3 be abelian groups. Prove that ?1 × ?2 × ?3 is an abelian group.
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Q: 2. Let G be a group. Show that Z(G) = NEG CG(x).
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Q: belong to a group. If |a| = 12, |b| = 22, and (a) N (b) # {e}, prove that a® = b'1.
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Q: 6. If G is a group and a is an element of G, show that C(a) = C(a')
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Q: 6. Prove the following groups are not cyclic : (a) Z x Z (b) Z6 × Z (c) (Q+, ·) (Here, Q+ = {q € Q…
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Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, CG(A) = A and…
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Q: Let S = {x €R | x + 3}. Define * on S by a * b = 12 - 3a - 3b + ab Prove that (S, *) is a group.
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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…
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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: 7 Suppose that a E G. For each y in the conjugacy class of a, let G(a y)= {g E G: gag= y} be the set…
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Q: Which of the following is not true? O The order of U(10) is 4 O U(10) is an torsion-free group O…
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Q: 64
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Q: F. Let a e G where G is a group. What shall you show to prove that a= q?
A: Solution: Given G is a group and a∈G is an element. Here a-1=q
Q: and are groups. Let (2, 9)| E A, y E. and denne the operation on X x Y as (T1, 41) * (x2, Y2) = (x1…
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Q: Label each of the following statements as either true or false. The Cayley table for a group will…
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Show that each of the following is not a group.
1. * defined on Z by a*b = |a+b|
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- 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. Let x,y, and z be elements of a group G. Then (xyz)1=x1y1z1.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.
- 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.Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)If a is an element of order m in a group G and ak=e, prove that m divides k.
- True or False Label each of the following statements as either true or false. A group may have more than one identity element.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. Two groups can be isomorphic even though their group operations are different.
- Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.