Which of the following groups are isomorphic? i) Z/75Z ii) Z/3Z ® Z/25Z iii) Z/15Z @ Z/5Z iv) Z/3Z @ Z/5Z® Z/5Z.

Q: Consider the group G = {x € R such that x + 0} under the binary operation x*y = -2xy The inverse…

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Q: Label the following statement as either true or false.   The alternating group A4 on 4 elements is…

A: We have to state whether the given statement is true or false : The given statement is : The…

Q: (d) Show that Theorem 1 does not hold for n = 1 and n = 2. That is, show that the multiplicative…

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Q: 6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D

A: Fact 1:In a group G, if x∈G such that xn = e, then O(x)|n (e→ identity element.)i.e. order of x…

Q: 3. Define an operation on G = R\{0} x R as follows: (a, b) (c,d) = (ac, bc + d) for all (a, b),…

A: 3. Define an operation * on G=ℝ\{0} ×ℝ as follows: (a,b)*(c,d)=(ac, bc+d) for all (a,b), (c,d) ∈G…

Q: Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that…

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Q: Let G be a group containing elements a and b. Express (ab)^2 without parentheses. Do not assume that…

A: Given that G is a group and let a,b belongs to G. The given expression is

Q: Define ∗ on ℚ+ by a ∗ b =ab/2 . Show that ⟨ℚ+,∗⟩ is a group.

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Q: 12. Prove that the following groups are not cyclic: (a) Z2 x Z2 (b) Z2 x Z (c) Z x Z.

A: To show the following groups are not cyclic. If a group is cyclic the there exit an element in that…

Q: Show that Z is not isomorphic to Zž by showing that the first group has an element of order 4 but…

A: The given groups: Z5x and Z8x That is, Z5x =Number of elements of a group of Z5 Z8x =Number of…

Q: Let U(n) be the group of units in Zn. If n > 2, prove that there is an element k E U(n) such that k2…

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Q: The group GLQ,R) abelian group is an

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Q: For any group elements a and x, prove that |xax-1| = |a|.

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Q: For any elements a and b from a group and any integer n, prove that(a-1ba)n = a-1bna.

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Q: (a) What does it mean for two groups to be isomorphic?

A: see my solution below

Q: List the six elements of GL(2, Z2). Show that this group is non-Abelian by finding two elements that…

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Q: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

A: Given :              A group of order 4  Corollary :                    The order of an element of a…

Q: What are the three things we need to show to prove that an ordered pair is a group?

A: We have to give the properties of an ordered pair to prove that it is a group.

Q: 5. Let p be a prime. Prove that the group (x, ylx' = yP = (xy)P = 1) is infinite if p > 2, but that…

A: The solution which makes use of matrix theory is presented in detail below.

Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

A: To identify the given group is cyclic or not.

Q: 1- Prove that if (Q - {0},) is a group, and H = 1+2n 1+2m 9 n, m e Z} is a subset of Q-{0}, then…

A: We need to prove that H=1+2n1+2m∋n,m∈Z is a subgroup of Q-0 A subset W of a group V is said to be a…

Q: In group theory (abstract algebra), is there a special name given either to the group, or the…

A: Yes, there is a special name given either to the group, or the elements themselves, if x2=e for all…

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

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Q: 5. Prove that the group (x, y|x = yP = (xy)P = 1) is infinite if %3D %3D n> 2 but that if n = 2 it…

A: To prove that the group x, y|xp=yp=xyp=1 is infinite if p>2, but that if p=2, it is a Klein…

Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

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Q: Consider the group G={x € R such that x#0} under the binary operation *: Th identity element of G is…

A: Solution: Since for any x,y∈G, the operation * is defined as x*y=-2xy The identity element is e=-12…

Q: Find the order of each of the elements of the group ((Z/8Z*, * ). Is this group cyclic? Do the same…

A: To investigate the orders of the elements in the given groups

Q: 3. Use the three Sylow Theorems to prove that no group of order 45 is simple.

A: Simple group: A group G is said to be simple group if it has no proper normal subgroup Note : A…

Q: ?Which of the following statement is true The largest order among the orders of all the cyclic…

A: We go through each options one by one to choose the correct.

Q: Consider the group G = {x € R]x # 1} under the binary operation : *• y = xy – x-y +2 The identity…

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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: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?

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Q: Show that if aEG, where G is a group and |a| = n then : %3D a' = a' if and only if n divides. -j

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Q: Show that if p and q are distinct primes, then the group ℤp × ℤq is isomorphic to the cyclic group…

A: We have to show that if p and q are distinct primes, then the group Zp×Zq is isomorphic to the…

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: Which of the following is nontrivial proper sub- group of Z4? {0, 2} O Diophantus of Alexandria {0,…

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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: The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of…

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Q: Find the group homomorphism between (Z, +) and (R- (0},.)

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Q: 2) Determine whether or not the groups Z10 × Z4 and Z, × Z20 are isomorphic. Justify your answer.

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Q: G, ba = ca implies b = c and ab = ac implies b = c for elements a, b, c E G. 31. Show that if a? = e…

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Q: 2. Are the groups Z/2Z x Z/12Z and Z/4Z x Z/6Z isomorphic? Why or why not?

A: Here we have to show that given groups are isomorphic

Q: ) Prove that Z × Z/((2,2)) is an infinite group but is not an infinite cyclic grou

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Q: 25. Prove that R* x R is a group under the operation defined by (a, b) * (c, d) = (ac, be + d).

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Q: 9. Show that the two groups (R', +) and (R' – {0}, -) are not isomorphic.

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Q: The set numbers Q and R under addition is a cyclic group. True or False then why

A: Solution