2. Show that in a group if x has inverse y and y and a right an inverse r, then y and r are the same element.

Q: Does every set A have an inverse? What is it? d) Is (U, A) a group?

A: c) Let A∈U. To find the inverse of A. For, let us find the identity element first. A△∅=A-∅∪∅-A=A∪∅=A…

Q: true or false, Let a and b be elements of a group G. If a = a −1 and b = b −1 , then ba is the…

A: Given a and b be elements of a group G. a = a−1 and b = b−1

Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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Q: Determine whether the following is a semigroup, monoid, or a group. 1. G=Z, a*b = a + ab 2. G=2Z,…

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Q: Let G be a group and let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is…

A: see below the answer

Q: In the group D4, let a = sp and b = sp3. The inverse of of ab is: a) sp2 b) ps c) p? d) None a C

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Q: If any two members P and Q of a group satisfy the relation P = Q'PQ, then P and Q commute. Select…

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Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.

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Q: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.

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Q: Let G be the group with presentation (x,y: 1² = 1, r² = y²₁ xy = y ¹). Decide how many elements are…

A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

Q: Let G be a group and let H< G. If [G: H] = 16 and |H| = 21, then what is |G|?

A: The expression, G:H can be written as GH .

Q: 2) (b*a)1 = . If a, b are elements of a group G?

A:

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: Let a and b belong to a group. If la| = 12, \b| = 22, and (a) N (b) + {e}, prove that a6 = bl1.

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Q: shiw that in a group if x has an inverse y abd a right inverse r, then y and r are the same element

A:

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

A: see my solution below

Q: 10. Given the group (S = {a,b,c, d},®) a a d ic a b a a d d The inverse of an element r, x E Sis D-…

A: Inverse element is an element which when operated with its inverse gives the identity

Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: 1. Assume (X, o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) |x € X,y E Y} and define the…

A:

Q: 2. Let G be a group. Pro-

A: Let G be a group .

Q: Let a and b belong to a group. If la] = 24 and |b| = 10, what are the possibilities for (a) n (b)i?

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Q: For group elements a, b, and c, express (ab)3 and (ab-2 c)-2 withoutparentheses.

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Q: The elements of order 8 in the group (Zg ,+) is a)4 b) 5 c)6 d) 8

A: In this question we have to find the number of elements of order 8. (a) 4 is correct option

Q: 3. Consider the group (Z,*) where a * b = a + b – 1. Is this group cyclic?

A: 3. Given the group ℤ,* where a*b=a+b-1. Then, 1*x=x*1=x+1-1=x Here 1 serves as the identity for Z.

Q: If X and Y are completely unrelated, r will be close to 1. Group of answer choices True False

A: If X and Y are completely unrelated, r will be close to 1. Group of answer choices True False

Q: Find groups that contain elements a and b such that |a|=|b|= 2 and a. |ab| =3, b. |ab| =4, c. |ab|=…

A: Given : |a|=|b|= 2 To find:  |ab| = ?

Q: Let E = Q(√2, √5). What is the order of the group Gal(E/Q)?What is the order of Gal(Q(√10)/Q)?

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Q: If a1, a2, . . . , an belong to a group, what is the inverse of a1a2 . . . an?

A:

Q: Suppose G is a group and r, be G so that r = b and r = b. Solve for a in terms of b.

A: Given: G is a group, and x,b∈G, so that x3=b5 and x8=b2. Formula used: Basic formula in power and…

Q: Consider the group D4

A: Given: Group D4=a,b=e=1,a,a2,a3,b,ab,a2b,a3b and a=1 2 3 4, b= 2, 4 To find : The value of…

Q: belong to a group. If |a| = 12, |b| = 22, and (a) N (b) # {e}, prove that a® = b'1.

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Q: Find the inverse of the element 9 in the group (U(10), ®10).

A:

Q: 6. If G is a group and a is an element of G, show that C(a) = C(a')

A:

Q: Assume (X,o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) | æ € X, y E Y} and define the…

A: The given question is related to group theory. Given: X , ∘ and Y , ∙ are groups.  Let X × Y = x,y |…

Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({a +bv2 : a,…

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Q: Q/ Let G = Z and a * b = a + b show that (G,* ) is a group or not.

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Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: any two elements x and y in a group G, there is a unique element z in G such that y = xz.

A:

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

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Q: Suppose that G = (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).

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Q: If b and c are the inverses of some element a in a group G then a) b =c b) b c b. c) b = kc for some…

A: Let a identity element e.. And b is inverse of a. Then a.b=e And if c is the inverse element of a…

Q: Given the group (S = {a,b,c, d},8) %3D a d a b a b d - d The inverse of an element z, z e Sis a a с…

A: The solution is given as

Q: Find the solution set for each of the following with the representation of the group on the number…

A: To find - Find the solution set for each of the following with the representation of the group on…

Q: In D4, the centralizer of the group at H is equal to?

A:

Q: Which of the following groups are cyclic? Justify. (a) G = U(10) = {k e Z10 : ged(k, 10) = 1} =…

A: We know that  1)Every cyclic group is almost countable  2) Every finite cyclic group is isomorphic…

Q: The inverse of 3 in the group (Z5, o5) is

A: Definition:

Q: and are groups. Let (2, 9)| E A, y E. and denne the operation on X x Y as (T1, 41) * (x2, Y2) = (x1…

A:

Q: Let X be a non-empty set, Show that whether (P(X),n) is a group or not

A: given  Let x be a non empty set, Show that whether  (p(x),∩) is a group or not we need to show it is…