Let R = R – {1} be the set of all real numbers except one. Define a binary operation on R by a * b = a + b + ab,Va, b, e R. Prove (R, *) is a group. Is it abelian? Is (Q, *) a subgroup? Is (2, *) a subgroup?

Q: . Let H be a subgroup of R*, the group of nonzero real numbers un- der multiplication. If R* C H C…

A: H be a subgroup of R*, the group of nonzero real numbers under multiplication. R+⊆ H ⊆ R*. To prove:…

Q: Show that if H and K are subgroups of an abelian group G, then {hk|h € H and k e K} is a subgroup of…

A: A set G is called a group if it satisfies four properties  Closure property:  ab∈G  where a,b ∈G…

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: Let G be the subgroup of GL3(Z₂) defined by the set 100 a 10 bc1 such that a, b, c Z₂. Show that G…

A: The given set of matrix is 100a10bc1 where a, b, c∈ℤ2. To find: the group to which the given set is…

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

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Q: Prove that for a fixed value of n, the set Un of all nth roots of 1 forms a group with respect to…

A: Fix an n∈ℕ. Let U=u∈ℂ | un=1 .  Note that U⊂ℂ* and ℂ* is a group under multiplication. Let u,v∈U…

Q: prove that the group G=[a b] with defining set of relations a^3=e, b^7=e, a^-1ba=b^8 , is a cyclic…

A: We need to prove that , group G = a , b  with defining sets of relations a3 = e   ,  b7 = e  also…

Q: Let G be a group with |G| = pq, where p and q are prime. Prove that every proper subgroup of G is…

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Q: integer 16. If EG→His a surjective homomorphism of groups and G is abelian, prove that H is abelian.…

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Q: 2. Let R = R – {1} be the set of all real numbers except one. Define a binary operation on R by a *…

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Q: Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)-1 = a-'b-1 %3D

A: Given: The statement is, let a and b be elements of a group G. Prove that G is abelian if and only…

Q: Let G be the subgroup of GL3(Z2) defined by the set 100 a 10 b C 1 that a, b, c Z₂. Show that G is…

A: Given: G is the subgroup of GL3ℤ2 which is defined by the set of matrix 100a10bc1 where a, b, c∈ℤ2 .…

Q: Find all distinct subgroups of the quaternion group Qs, where Q8 = {+1,±i,±j, £k} Deduce that all…

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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: * O Abelian O…

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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: Let G be a group with IG|=187 then every proper subgroup of G is:* O Cyclic None of these Non cyclic…

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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 G be an abelian group of order 2n, where n is odd. Use Lagrange's Theorem to prove that G…

A: Given :               The group G, which is an abelian group of order 2n, where n is odd…

Q: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

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: 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 G be a group with the order of G = pq, where p and q are prime. Prove that every proper subgroup…

A: Consider the provided question, Let G be a group with the order of G = pq, where p and q are prime.…

Q: Let G be a group of finite order n. Prove that an = e for all a in G.

A: Let G be a group of finite order n with identity e.           Since G is of finite order…

Q: Let M be an abelian group and D a subgroup. For any element m of M, where m has order 2, define…

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Q: Let (G1, +) and (G2, +) be two subgroups of (R, +) so that Z+ ⊆ G1 ∩ G2. If φ : G1 → G2 is a group…

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Q: 6. Prove that if G is a group of order 231 and H€ Syl₁1(G), then H≤ Z(G). n Core

A: Given that,  G is group of order 231 and H∈syl11G. We first claim that there is a unique Sylow…

Q: Let S = R\{-1} and define a binary operation on S by a * b = a +b+ ab. Prove that S is an abelian…

A: To show that S is an abelian group, we have to prove all these properties 1) S is closed under…

Q: Prove that every subgroup of Z is either the trivial group, {0}, or nZ = {nx | x E Z} for some n E…

A: To prove: That every subgroup of ℤ is either the trivial group{0} or nℤ=nxx∈ℤfor some n∈ℕ. Proof:…

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: Let H be the set of elements (ª of GL(2, R) such that ad– bc=1. Show that H is a subgroup of GL(2,…

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Q: Let p : G → G' be a group homomorphism. (a) If H < G, prove that 4(H) is a subgroup of G' (b) If H <…

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Q: Let G be a group, and N ⊆ Z(G) be a subgroup of the center of G, Z(G). If G/N, the quotient group is…

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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. Define a new binary operation * on G by the formula a * b = b · a for all a, b…

A: We proved (G,*) is a group if it satisfied the following axioms.

Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

Q: '. Assume that G is a group such that for all x E G, x * x = e. Prove that G is an abelian group.

A: Consider any two elements a and b in G. So, a,b,ab,ba∈G. Note that I am directly writing the…

Q: Let G be an abelian group, then (acba)(abc)¯1 is

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Q: If A is an abelian group with A <G and B is any subgroup of G, prove that ANB < AB.

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Q: (a) Give the definition of a gyclic group. (b) Prove that every eyclic group is abelian . (c) Prove…

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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: 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 R be he set of real mumbers and let bER %3D 1 Show that G is abelian group an under…

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Q: Let m and n be integers that are greater than 1. (a) If m and n are relatively prime, prove that Zm…

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