Let S = {x €R | x + 3}. Define * on S by a * b = 12 - 3a - 3b + ab Prove that (S, *) is a group.

Q: Show that in a group G of odd order, the equation x2 =a has aunique solution for all a in G.

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Q: Let m and n be relatively prime positive integers. Prove that the order of the element (1, 1) of the…

A: To prove ℤm×ℤn≅ℤmn ⇔ gcd(m,n)=1 Part 1) Proof of 'only if'  part  ℤm×ℤn≅ℤmn ⇒ℤm×ℤn is cyclic. Let…

Q: Let R = R\ {-1} and define the operation ♡ on R by a♡b = ab + a +b Va, be R. Show that (a) V is a…

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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: 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: If a is a group element, prove that every element in cl(a) has thesame order as a.

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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: 3. Let n eN be given. Is the set U = {A: det A = ±1} C Matnxn(R) a group under matrix multipli- %3D…

A: By using properties of group we solve the question no. 3 as follows :

Q: Let G be a group. Show that for all a, b E G, (ab)2 = a2b2 G is abelian

A: To prove that the group G is commutative (abelian) under the given conditions

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 each of the following is not a group. 1. * defined on Z by a*b = |a+b|

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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: Let G be a group, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

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

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Q: consider the Set. H=E34+5m nime Z 27-1Is. idi group of Z Su b

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

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Q: Prove that in a group, (a-1)-1 = a for all a.

A: By definition (a-1)-1=a are both elements of a-1. Since in a group each element has a unique…

Q: Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

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Q: Let C be a group with |C| = 44. Prove that Cmust contain an element of order 2.

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Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

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Q: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

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Q: Prove that in a group, (a-1)¯' = a for all a.

A: To prove that in a group (a-1 )-1=a for all a.

Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

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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: Let Dg be the Dihedral group of order 8. Prove that Aut(D8) = D8.

A: We have to solve given problem:

Q: Prove that (ab)2 = a²b² for all a, b in a group G if and only if G is Abelian.

A: Let G be set and "·" be a binary operation then G,· is a group if  Clouser property. That is, if…

Q: If (G, * ) is a group with a a for all a in G then G is abelian

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Q: Prove that An even permutation is group w.y.t compostin Compostin function.

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Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

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Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

Q: Q:: (A) Prove that 1. There is no simple group of order 200.

A: A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group…

Q: 24, Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then a and b are…

A: Given:  Let G be a group. ZG be its center.  We know that ZG=z∈G: ∀g∈G,zg=gz ....i First we will…

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: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ 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: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: Define * on Q by a +b= qb Is Q a group under *? 210

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Q: Determine whether an onto homomorphism between the groups D6 and D3 + Z2 exists.

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Q: If I is an ideal of R, then by definition, (I, +) is an abelian group. Consequently, it has an…

A: Since I is ideal , therefore I is definitely a subset of the ring R.

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: Let G = {a + b/2|a, b € Z}. Show that G is a group under ordinary addition.

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Q: Prove that is (ab)-1 = a-1b-1 for all a,b in group G, then G is abelian.

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

A: NOTE:  We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: 64

A: Under the given conditions, to show that the cyclic groups generated by a and b have only common…

Q: Find the order of each element of the group Z/12Z under addition

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Q: a. Show that (Q\{0}, + ) is an abelian (commutative) group where is defined as a•b= ab b. Find all…

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Q: Let G be a group and suppose that (ab)2 = a²b² for all a and b in G. Prove that G is an abelian…

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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 necessarily:…

A: The answer is given as follows :

Q: Every abelian cy elic O True O False group is

A: to check  whether every abelian group is cyclic or not? proof let a euler group U8=1,3,5,7 let…