1.) Let G={1,−1,i,−i} be the group of 4 complex numbers under multiplication.  (a) Is {1,i} a subgroup of G and why?

Q: 1- Let (C,) be a group of non-zero complex number and let H = {x + iy}| x² + y2 = 2}. Then (H,;) is…

A: Since you have asked multiple question, we will solve first question for you. If you want any…

Q: Prove C*, the group of nonzero complex numbers under multiplication, has a cyclic subgroup of order…

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Q: A. Determine whether the given subset of the complex numbers is a subgroup of the of complex numbers…

A: Given

Q: Theorem :- Let (G,-) be a group then :- 1- (Hom(G),) is semi group with identity. 2- (A(G),) is…

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Q: Prove that The set of all complex number is a group under multiplication, and G={1,-1,i,-i}  is a…

A: Hi! You have posted multiple questions. As per norms, we will be answering only one question and as…

Q: (a) Explain why it is impossible for any set of (real or complex) numbers which contains both 0 and…

A: To solve the given problem, we use the defination of group.

Q: Theorem 3-12 Every infinite cyclic group is isomorphic to (Z,+). Proof.

A: Explanation of the solution is given below...

Q: 3. Prove that the groups (R.) and (R,+) are not isomorphic, where (R* R{0},-) is the multiplicative…

A: We have to show that the groups ℝ, + and ℝ*, · are not isomorphic. Here, the group ℝ*, · does not…

Q: Let G ={1,−1, i,−i} be the group of 4 complex numbers under multiplication.  (a) Is {1, i} a…

A: In the given question we have to determine if given set is subgroup of G or not.

Q: 1.) Let G₁ = £1₁ 1₁ i -i) be group of 4 Complex numbers under mutiplication. a) is £1, i} a subgroup…

A: Given group under multiplicationG = 1, -1,  i , -i  A subset H of G will be subgroup of G if H…

Q: Prove C*, the group of nonzero complex numbers under multiplication, has a cyclic subgroup of order…

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Q: Here is the question I'm needing help with.  Prove C*, the group of nonzero complex numbers under…

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Q: 2. Let R = R – {1} be the set of all real numbers except one. Define a binary operation on R by a *…

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Q: List the elements of the , i. e. cyclic subgroup generated by i of the group C* of nonzero complex…

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Q: Exercises: 1- Let (C,) be a group of non-zero complex number and let H = {x + iy}| x² + y? = 2}.…

A: Subgroup of a group

Q: (1) Compute the fundamental group of X = R³\{x – axis}U {y – axis} U{z – axis}.

A:

Q: Prove that the group of positive rational numbers, Q+, under multiplication is not cyclic.

A: Group under addition cyclic or non cyclic

Q: There is a unique group law on the 8- element set Q = {±1, ±i, ±j, ±k} that makes i2 = j2 = k2 = ijk…

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Q: Example: H.W 1- Let (C\{0},.) be a group of non-zero complex number and let H = {a + ib, a? + b2 =…

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Q: .) Prove that for every element a of a Group G, Z(G) is a subset of C(a)

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Q: Show that 1 y? ry under a Lie group corresponding to the symmetry generator X = r² a + ry

A: Given,

Q: 1. 4. Let a be a complex number such that lal > 2. Prove that 1. and a 1 generate a free group.

A: 4 Given abe a complex number such that a≥2. We have to prove that 10a1 and 1a01 generate a free…

Q: 1.) Let G ={1,−1,i,−i} be the group of 4 complex numbers under multiplication.  (b) Is {1,−1} a…

A: Use Finite subgroup test : Let H  be  a non empty finite subset of a  group G. If H is closed under…

Q: Determine the behaviour of the set of 4 elements {1,-1,i,-i} under the ordinary multiplication of…

A: Given, The set of 4 elements are 1,-1,i,-i.It is to be shown that this set is group.

Q: QUESTION 9 Find up to isomorphism all Abelian groups of order 36.

A: We have to find the all abelian groups of order 36 up to isomorphism.

Q: Find an isomorphism o from the multiplicative group G of nonzero complex numbers to the…

A: We need to find ϕ for the multiplicative group G of non zero complex numbers to the multiplicative…

Q: 2. Prove that a free group of rank > 1 has trivial center.

A: Given:Prove that a free group of rank>1 has trivial center

Q: Using the Theorem of Lagrange, prove that a group G of order 9 is abelian

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Q: 5. Recall that i = V-I and let G = {1,a, B} where a = the operation on multiplication. remember that…

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Q: Prove: (R+) (Q++) (Rx) ) X) all are non-cyclic group ?

A: Cyclic Group: A group G is called cyclic if there is an element a in G such that  G=a=an| n∈Z, where…

Q: List the elements of the (i) , i. e. cyclic subgroup generated by i of the group C* of nonzero…

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Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.

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Q: Prove C*, the group of nonzero complex numbers under multipliation, has a cyclic subgroup of order n…

A: n th root of unity is the cyclic subgroup of order n.

Q: 12. Let "- (a + bi |a.bER, ab a 0}. Determine whether H is a subgroup of the complex numbers C with…

A: 12. Consider the group of complex numbers C with addition. Let H=a+bi|a,b∈R, ab≥0. Clearly, the…

Q: Prove that group A4 has no subgroups of order

A: Topic- sets

Q: (7) Define GL2 (R) to be the group of invertible 2 x 2 matric manifold, cc this group has the…

A: Define GL2R  to be the group of invertible 2×2 matrices.  To prove that this group has the structure…

Q: 3. Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the…

A: Result: Let G, * be a group and H is a subset. The subset H is said to be subgroup, if for every a,…

Q: Let G = {2'3m5n : l, m, n E Z}. G is a group of rational numbers under the usual multiplication.…

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Q: 1- Let (C\{0},.) be a group of non-zero complex number and let H = {a + ib, a² + b² = 1} then (H,.)…

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Q: Prove that (Z × Z)/((0,1)) is an infinite cyclic group. Prove that (Z × Z)/((1,1)) is an infinite…

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Q: Let G ={1,-1, i, -i} be the group of 4 complex numbers under multiplication. (b) Is {1,–1} a…

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Q: show that under complex multiplication, G={1,-1.i,-i} is an abelian group?

A: we have proved this by cayley table.

Q: Show that the mapping Φ(a + bi) = a - bi is an automorphism of the group of complex numbers under…

A: Consider the equation

Q: Write out the Cayley table for the group {1, -1, i, -i} under multiplication where i = sqrt(-1). (In…

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Q: Find all finite-dimensional complex representations of the group Z.

A: To find all finite-dimensional complex representation of the group Z.

Q: List all of the elements in each of the following subgroups. (1) The subgroup of Z generated by 7…

A: Hey, thanks for your question. According to our policy I m answering first three subparts . Please…

Q: QUESTION 14 Find up to isomorphism all Abelian groups of order 18.

A: To find- Find up to isomorphism all Abelian groups of order 18.

Q: OLet a and b be elements of a group G. Prove that G is abelian if and only if (ab)- = a¯\b-!.

A: Prove that G is abelian if and only if (ab)-1=a-1b-1. For all a and b be elements of a group G.

Q: 1- Let (C,) be a group of non-zero complex number and let H = {x + iy}| x² + y² = 2}. Then (H,) is a…

A: “Since you have asked multiple question, we will solve the first question for you. If you  want any…

Q: (4) Find the Galois group of the polynomial r + 1.

A: Since you have asked multiple question, we will solve any one question for you. If you want any…