(14) Let G be a finite cyclic group of order n with identity element e. Prove that if a € G, then cd = e for some divisor d of n.

Q: , Let G be a finite Abelian group of order n. Suppose G = {91,92,.. {91, 92, 9n} and let x be an…

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Q: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: 6. Apply Burnside's formula to compute the number of orbits for the cyclic group G = {(1,5) o (2, 4,…

A: Solve

Q: Let m, n e Zt such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: We have M and N are abelian groups of order m and n. so, we know that MN is abelian group.

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: 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: Let X,Y be cyclic groups of order 2,3 respectively.  Find the orders of the elements of X×Y.

A: Let X, Y be cyclic groups of order 2,3 respectively then we have to find the   the orders of the…

Q: Any subgroup of F2 (the free group upon 2 generators) is not abelian." Is this statement true or…

A: The given statement is Any subgroup of F2 (the free group upon 2 generators) is not abelian.

Q: Prove that the group G = [a, b

A: Given, the group G=a, b with the defining set of relations…

Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D

A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn

Q: | Let H = U(35) = {[m] =3z|gcd(m, d) = 1}. Compute ord(6) 35Z and ord(-1) in H. Show that H is not a…

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Q: Let G (a) be a cyclic group of order n. For each integer m, define a map fm : G -> G by fm(x) т =…

A: Let G=<a> be a cyclic group of order n. for any integer m defines:

Q: (a) Show that a group G is abelian, if (ab)² = a²b², for a, b € G[C.

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Q: 1. Prove or disprove (a). The order of the element (2, 3) € Z6X Z15 is 5. (b). The group Z7X Z17 x…

A: As per the guidelines, I am supposed to do the first three sub parts. Kindly post the other…

Q: b. Find all abelian groups, up to isomorphism, of order 360.

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Q: 4.6. (a) Let G be a cyclic group of order 6. How many of its elements generate G? Answer the same…

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Q: 1. Let G be a cyclic group of order n and k❘n. Show that G has a subgroup of order k; "

A: To prove that a cyclic group ( G ) of order ( n ) has a subgroup of order ( k ) for every ( k ) that…

Q: Let G be a finite abelian group

A: Let d = gcd(m; n).We shall use induction on d.Assume  first that d = 1.

Q: (a) G = Z where a * b = a + b – 1 for all a, b EG (b) G = {r € Q |r > 0 } where a * b = ab for all…

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Q: Subgroup of G. (b) Let G be an abelian group. Let T = {ge G: g = e for some positive integer n}.…

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Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = –S, |S| = 3 and (S) =…

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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 the statement: Let g be a group, if g is abelian then (ab)2 =  a2b2.

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Q: Let G be a group of order pq, where p and q are primes. Prove that either G is abelian or Z(G) =…

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Q: 8. Let G be a finite abelian group. (a) Let d be a positive divisor of |G|. Prove that G has a…

A: a) We need to prove that if d is a positive divisor of |G| then G has a subgroup of order d. Let the…

Q: b. Show that a cyclic group of order 6 has a cyclic group of order 2 as a homomorphic image.

A: To show that a cyclic group of order 6 has a cyclic group of order 2 as a homomorphic image, we need…

Q: 4. Let G be a group of order 49. Prove that G is cyclic or g7 = e for all g € G.

A: Given a group of order 49, then to prove that either G is cyclic or g7=e for all g∈G.

Q: Let G= be a finite cyclic group of order n. Suppose that n = dq where d and q are non-negative…

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Q: 6. Let G be GL(2, R), the general linear group of order 2 over R under multiplication. List the…

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Q: Let G be a group with IG|=209 then every proper subgroup of G is: * O Non abelian O Non cyclic O…

A: See the detailed solution below.

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

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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: Suppose x is an element of a cyclic group of order 15 and x3 = x7 = x°. Determine |x13].

A: According to a theorem in group theory , If G is a finite group and a∈G be an element in the group…

Q: Let a, b be elements of an abelian group G and let m = |a| and n = |b|. If m and n are relatively…

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Q: Show that x is a generator of the cyclic group (Z3[x]/<x3 + 2x + 1>*.

A: Given: The cyclic group Z3x/x3+2x+1*We need to show that x is a generator of the given cyclic group.

Q: If (G, *) is an infinite cyclic group, then (G, *) (Z, +). Select one: O True O False

A: Given the statement if G, * is an infinite cyclic group then G, *≅ℤ, +. To determine whether the…

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: 3. Let G be a group with identity e. Show that if a ★a=e for all a in G, then G is Abelian.

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Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Show that x is a generator of the cyclic group (Z3[x]/<x3 + 2x + 1>)*.

A: In this question, we have to show that x is a generator of the cyclic group  ℤ3xx3+2x+1*. We know…