f G is a finite group and Z is its centre, then class equation of G is expressi o(G) o(G) = o(Z) + E o(N(a)) a & Z where the summation runs over one element a in each conjugate class.

Q: Define f: G→ G', where f(x) = (3x for every x € G. the f homorphism. Determine the f

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Q: 2. Let G = (R*, .), the multiplicative group of nonzero real numbers. Let H = {x ER: x -1} be the…

A: 2. Let , the multiplicative group of non-zero real numbers.Let  be the group with operation .Prove…

Q: If (G,X) be the group of all real 2 x 2 matrices (b) such that ad - bc #0 with matrix multiplication…

A: Given if G, × be the group of all 2×2 matrices abcd such that ad - dc≠0 with matrix multiplication…

Q: The set of matrices S = # x€ R forms a group under multiplication operation with identity element…

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Q: 12. Describe the group of the polynomial (x* – 5x? + 6) € Q[x] o over Q.

A: please see the next step for solution

Q: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy<minox, oy. Let us take the group…

Q: If f is a homomorphism of a group G into a group G' then (i) ƒ(e) = e', where e is the identity of G…

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Q: (a) (i) Let G = R-{0} × R. Prove that (G, *) is a group where is defined by (a, b) * (c, d) = (ac,…

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

Q: Let A and B be rings. Define addition and multiplication on the Cartesian product A x B by…

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Q: means a cyclic group but 3+i can't turn into 3-i with a positive integer exponent.

A: Please see the answer in next step.

Q: Let x= then find the transformation R (x). 33-2 2. -3-23 2 None of these 3/3-2 -3-33 3-23 2 -2-33

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Q: 9. Classify the following groups in the sense of the Fundamental Theorem of Finitely Gener- ated…

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Q: (B) Let G be a set of all real numbers excep: . Define * on G by a * b = |a|b, where |a| is the…

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Q: There are four questions about this problem in the test shown in random order. Out of 1000 customers…

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Q: f G is a finite group and Z is its centre, then class equation of G is expressible as D(6) 0(N(a))…

A: G is a finite group and Z is its centre, then we have to show that the  class equation of G is…

Q: Example(6-3): For each r ER, define pr: (R,+) → (R,+) by pr(a): = ar for each a ER Show that p, be…

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Q: Consider the group of nonzero real numbers, R* = R \ {0}, under mul- tiplication and the group of…

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Q: (a) Explain why any automorphism (b) Show that any automorphism that Aut (S3) 6. of S3 takes a…

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Q: 5 Let G be the set of all functions f: R → R of the form f(x) = ax + b where a and b are real…

A: Introduction: Function composition is an operation where two functions are combined to form a new…

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: Show that the groups (Z/4, +4) and (Z/5 – {[0]}, x5) are isomorphic.

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Q: Suppose that f1x2 is a fifth-degree polynomial that is irreducibleover Z2. Prove that every…

A: It is given that,  f(x) is a fifth degree polynomial that is irreducible over ℤ2. We have to show…

Q: Consider the group G (x E R]x 1} under the binary operation : ** y = xy-x-y +2 If x E G, then x =…

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Q: Show that there are no simple groups of order 255 = (3)(5)(17).

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Q: Find a particular solution of the equation y4 -y3 - y" - y' - 2y = 8x^5:

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Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a = e O a^4…

A: Given that f:G→G be a function such that fx=axa

Q: Let G = Z4 x Z6. Compute the factor groups G/((2, 3)) and G/((3, 3)). (In each case, write the…

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Q: (2) (Z x Z)/(6Z × 14Z)

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Q: (4) Let G be a finite abelian group. Let V {0} be an irreducible CG-module of finite degree. Let…

A: Given: In this problem, G is an finite abelian group. V≠{0} is an irreducible ℂG -module of finite…

Q: Show that x is a generator of the cyclic group (Z3[x]/<x3 + 2x + 1>*.

A: Given: The cyclic group Z3x/x3+2x+1*We need to show that x is a generator of the given cyclic group.

Q: If (G,X) be the group of all real 2 x 2 matrices (“ ") such that ad - be + 0 with matrix…

A: First we have to show that f:G→G' is homomorphism or not. To show homomorphism, we have to show…

Q: f: (A, SA)(B,S) be an order isomorphism and, Sa={xEA: x<naf## and Spa) =y eB= y <gf(@)}. 3) Let %3D…

A: Consider the given information: Let f:A,≤A⇒B,≤B be an order isomorphism. Sa=x∈A: x&lt;Aa≠∅ and…

Q: G is a finite group and f be an automonphism of G satisfying the conditian f(x)= x → x=e for xEG.…

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Q: Suppose f: A→ B is a non-zero group homomorphism. Which of the following are true? (i): If A = Z36,…

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