(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 = Z8 x Z6, and consider the subgroups H = {(0, 0), (4, 0), (0, 3), (4,3)} and K = ((2, 2)).…

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Q: Show that Z12 is not isomorphic to Z2 ⊕ Z6. Recall that ℤ? denotes the abelian cyclic group of order…

A: It has to be proved that Z12 is not isomorphic to Z2 ⊕ Z6. Isomorphism is defined as one-to-one…

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: Show that Z is not isomorphic to Zž by showing that the first group has an element of order 4 but…

A: The given groups: Z5x and Z8x That is, Z5x =Number of elements of a group of Z5 Z8x =Number of…

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 an Abelian group of order 16. Suppose that there are elementsa and b in G such that |a| =…

A: Given: G be an Abelian group of order 16 and a=b=4.To find: The isomorphism class of G.

Q: Can you solve in terms of y(t) = ?

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

A: Given that G is a group of order pq , where p and q are primes.Z(G) We have to show that G is…

Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

Q: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

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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: Define a binary operation on R\ {0}by x*y = . 2 Prove that this set with this binary operation is an…

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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: 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: (6) Let G be a noncommutative group. Show that G has a nontrivial subgroup.

A: We recal Cauchy's theorem and then use it

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: 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: An element x of a group satisfies x2 = e precisely when x = x-1. Use this observation to show how…

A: So no. of the elements of order 2 has to be odd.Explanation:Step 1: a = at ein ide">

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…