а H be the set of all matrices in GL2(R) of the form b. a
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Q: group ⟨a,b,c|b^4 = a,c^(−1) = b⟩ is abelian.
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Q: 3. Let n eN be given. Is the set U = {A: det A = ±1} C Matnxn(R) a group under matrix multipli- %3D…
A: By using properties of group we solve the question no. 3 as follows :
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Q: 3. You have already proved that GL(2, R) = {[ª la, b, c, d e R and ad – bc ± 0} forms a group under…
A: Note: There are two questions and I will answer the first question. So, please send the other…
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Q: ] Given the set S:= {2"5" : m, n e Z}. Does the set S together with as. multiplication form a group?…
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Q: QUESTION 5 Show that ifevery element in a group G is equal to its own inverse, then G is Abelian.
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A: As per our guidelines, I can answer only one question.
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- 27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.1.Prove part of Theorem . Theorem 3.4: Properties of Group Elements Let be a group with respect to a binary operation that is written as multiplication. The identity element in is unique. For each, the inverse in is unique. For each . Reverse order law: For any and in ,. Cancellation laws: If and are in , then either of the equations or implies that .Prove part e of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.
- True or False Label each of the following statements as either true or false. Let H1,H2 be finite groups of an abelian group G. Then | H1H2 |=| H1 |+| H2 |.Rework exercise 9 with G=GL(2,), the general linear group of order 2 over , and G= under addition. M2() Exercise 9: Let G be the additive group of 22 matrices over and G the additive group of real numbers. Define :GG by ([ abcd ])=a+d. (This mapping is called the trace of the matrix.) Prove or disprove that is a homomorphism. If is a homomorphism, find ker and decide whether is an epimorphism or a monomorphism.Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
- 31. a. Prove Theorem : The center of a group is an abelian subgroup of. b. Prove Theorem : Let be an element of a group .the centralizer of in is subgroup of.In Example 3, the group S(A) is nonabelian where A={ 1,2,3 }. Exhibit a set A such that S(A) is abelian. Example 3. We shall take A={ 1,2,3 } and obtain an explicit example of S(A). In order to define an element f of S(A), we need to specify f(1), f(2), and f(3). There are three possible choices for f(1). Since f is to be bijective, there are two choices for f(2) after f(1) has been designated, and then only once choice for f(3). Hence there are 3!=321 different mappings f in S(A).40. Prove or disprove that the set in Exercise is a group with respect to addition. 38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.
- 6. Let be , the general linear group of order over under multiplication. List the elements of the subgroup of for the given, and give. a. b.Exercises In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .