Give an example of a group G and elements x, y E G with o(xy) < min(o(x), o(y)). Give an example of a group G and elements x,y E G with o(xy) > o(x)o(y).

Q: (G, .) a group such that a.a = e for all a EG.Show that G is an abelian group. Let be

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Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

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Q: Explicitly construct the Galois group for r- 4r² + 2 over Q. To which group is this isomorphic?

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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: 2- Let (C\{0},.) be the group of non-zero -complex number and let H = { 1,-1, i,-i} prove that (H,.)…

A: To Determine: prove that H,. is a subgroup of a group of non zero complex number under…

Q: Consider the square X = [-1,1]2 = {(x, y)|x > -1, y < 1} and 0 = (0,0). Show that the fundamental…

A: image is attached

Q: If x is an element of a cyclic group of order 15 and exactly two of x3, x5, and x9 are equal,…

A: Given: The order of group is 15

Q: [Zp-(0),.] Where p is prime is an abelian group

A: We have to show that [Zp-(0),.] Where p is prime is an abelian group

Q: 6. Let G be a group with the following property: "If a, b, and c belong to G and ab=ca, then b = c."…

A: For a group to be abelian it should follow that ab=ba for all a,b belonging to G. Now we will…

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

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Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

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Q: Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

A: Consider the given information, Let G be a group of odd order then, |G|=2k+1 where k belongs to…

Q: Every group of order 4 is cyclic. True or False then why

A: Solution

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: implies y = z. Prove that G is Abelian. Prove that a group G is Abelian if and only if (gh)-1 = g¬…

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Q: %3D Let x belong to a group. If x² +e while x° = e, prove that about the order of r?

A: Given that x2≠e and x6=e To prove that x4≠e and x5≠e Suppose that x4=e  also x6=e therefore…

Q: Prove that the group G with generators x, y, z and relations z' = z?, x² = x², y* = y? has order 1.

A: In order to solve this question we need to make the set of group G by finding x, y and z.

Q: True or false? The group S3 under function composition ◦ is not a cyclic group

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Q: Q2: If G = R- {0} and a * b = 4ab ,show that (G,*) forms a commutative group? %3D

A: To show for the commutative group of (G, *), we verify the following properties of the commutative…

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

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Q: 6. Prove that if G is a group of order 231 and H€ Syl₁(G), then H≤ Z(G). me

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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: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion

A: According to the given conditions:

Q: 46. Determine whether (Z, - {0},6 ) is it a group or not? Explain your answer?

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Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

Q: Determine the order of (Z ⨁ Z)/&lt;(2, 2)&gt;. Is the group cyclic?

A: Given, the group                 We have to find the order of the group and also check, This is a…

Q: (2) (Z x Z)/(6Z × 14Z)

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Q: Q3: Describe the quotient group of a- (²/z, ·+) b- (2/z,+)

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Q: 2) Prove that Zm × Zn is a cyclic group if and only if gcd(m, n) cyclic group Z; x Z4. = 1. Find all…

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Q: Prove that the alternating group is a group with respect to the composition of functions?

A: Sn is the set of all permutations of elements from 1,2,.....,n which is known as the symmetric group…

Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

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

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Q: Show that Z12 is not isomorphic to Z2 ⊕ Z6.  ℤn denotes the abelian cyclic group of order n. Justify…

A: To show : ℤ12 is not isomorphic to ℤ2⊕ℤ6 Pre-requisite : P1. A group G is said to be cyclic if there…

Q: 5. Prove that the cyclic group Z/15Z is isomorphic to the product group Z/3Z × Z/5Z.

A: Definitions: Isomorphism: A mapping between two sets is called an isomorphism if it is one-to-one,…

Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

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Q: If G is a non-abelian group of order 8 with Z(G) {e}, prove that |Z(G)| = 2.

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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: Let G be a group. V a, b, c d and x in G, if axb cxd then ab = cd then G is necessa: Abelian Of…

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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: Decide if the abelian group Z/2 × Z/2 is cyclic or not. Prove your answer

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Q: 4. Consider the additive group Z. Z Prove that nZ Zn for any neZ+.

A: We know that a group G is said to a cyclic group if there exists an element x of the group G such…

Q: Let x, y be elements in a group G. Prove that x^(−1). y^n. x = (x^(−1).yx)^n for all n ∈ Z.

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Q: Let x belong to a group. If x2e while x : x + e and x + e. What can we say about the order of x? =…

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Q: .A group (M,*) is said to be abelian if إختر أحد الخيارات (x+y)=(y+x) .a O (y*x)=(x+y) .b O…

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Q: . Prove that the group Zm × Zn is cyclic and isomorphic to Zmn if and only if (m, n) = 1.

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Q: 1. Consider the groups (R+, ) and (R,+). Then R* and R are isomorphic under the mapping $(x) = log10…

A: We use the definition of cosets, isomorphisms to answer these questions. The detailed answer well…

Q: Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x…

A: Let X,◊ and Y,* are two groups. The Cartesian product of X and Y defined by X×Y=x,y | x∈X and  y∈Y.…