Let a and b be elements of a group with identity 1. Suppose |a| and |b| are relatively prime. Use Lagrange’s Thm. to prove that n ={1}.

Q: In the group Z24, let H =(4) and N= (6). (a) State the Second Isomorphism Theorem. (b) List the…

A: As per our guidelines only first three subquestions are solved. To get solution of remaining…

Q: For each of the following groups G, find an external direct product of cyclic groups of prime power…

A: We shall answer first three sub-parts only as per company guidelines. For others kindly post again…

Q: Let G be a group. Using only the definition of a group, prove that for each a E G, its inverse is…

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Q: 1. Let G be an abelian group with the identity element e. If H = {x²|x € G} and K = {x € G|x² = e},…

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

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Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

A: We have given a map ,  ϕ : A → B , where A = ℝ , +   ,  B = ℝ* , · such that  , ϕx = 2x We know that…

Q: let G be an abelian Group and let H and K be a susb group of G prove that HK ={hk / h an element H…

A: Consider the provided question, Your question is missing the last statement; I think you want to…

Q: For any elements a and b from a group and any integer n, prove that (a21ba)n 5 a21bna.

A: For any elements a and b form a group and any integer n, We have to prove that a-1ban=a-1bna.

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 G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: Every cyclic group or order n is isomorphic to (Zn, +n) and every infinite cycle group is isomorphic…

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Q: For any group elements a and x, prove that |xax-1| = |a|.

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Q: Let X be a group and we then let x and y be an element of X.  Prove that (x*y)^ -1 = a^-1 * b^-1 iff…

A: Since there are some mistakes in given typed question.question may like "Let X be a group and let…

Q: 1. Let G.*) be a group and a EG Suppose that a*a = a Prove or disprove that a must be the identity…

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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: for the cyclic group of order 16 generated by a , show that a^4 generates an invariant subgroup

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Q: Let G be a finite group with no element of order 3, and let a E G. (a) Can 3 divide o(a)? Either…

A: Given that G is a finite group such that ∀a∈G, o(a)≠3 (a) We have to determine whether 3 divides…

Q: Suppose that the fundamental group of X is Z and p(xo) is finite. Find the fundamental group of X.

A: Given fundamental group of X is  ℤ and p-1(xo) = finite value Now we have to find the fundamental…

Q: et G be a group with order n, with n > 2. Prove that G has an element of prime order.

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Q: Let G be a group and let a e G. In the special case when A = {a}, we write Cg(a) instead of Cg({a})…

A: The given problem is related with group theory. G is a group and a ∈ G. Here, we write Cga instead…

Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

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Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.

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Q: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a = a-1.

A: Let G be a group with respect to * . Let e be the identity element,and a is non identity element.…

Q: . Let G be the additive group Rx R and H = {(x,x) : x E R} be a subgroup of G. Give a geometric…

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Q: Let G be a group and let a e G. In the special case when A= {a},we write Cda) instead of CG({a}) for…

A: Consider the provided question, According to you we have to solve only question (3). (3)

Q: In group theory (abstract algebra), is there a special name given either to the group, or the…

A: Yes, there is a special name given either to the group, or the elements themselves, if x2=e for all…

Q: Use the Cayley table of the dihedral group D3 to determine the left AND right cosets of H={R0,F}.…

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Q: An element a of a group G has order n E z+ if and only if a" = e.

A: The given statement is False.

Q: Let G be a group, and let H < G. Assume that the number of elements in H is half of the number of…

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Q: Let G be a group and let a e G have order pk for some prime p, where k ≥ 1. Prove that if there is x…

A: As per the policy, we are solving first question. Please repost it and specify which question is to…

Q: 1. Let (G. *) be a group and a E G. Suppose that a *a = a · Prove or disprove that a must be the…

A: We have to solve given problem:

Q: Given that A and B is a group. Find out if þ: A→B is a homomorphism. If it is a homomorphism, also…

A: Group homomorphism is nothing but a function defined between two groups. The function must be closed…

Q: Consider the group G = Dg × Z/6Z, and let N = Dg x {0}. Show that N 4 G and that G/N = Z/6Z. Now…

A: Given is a group G and a subgroup N of G. First we use the definition of Normal subgroup to show…

Q: Let (G, -) be an abelian group with identity element e Let H = {a E G| a · a · a·a = e} Prove that H…

A: To show H is subgroup of G, we have show identity, closure and inverse property for H.

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: 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: Let p and q be distinct prime numbers and set n = pq. Find the number of generators of the cyclic…

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Q: Let G be a group with identity element e, and let H and K be subgroups of G. Assume that (i) H and K…

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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: 5. Let G be a group and let a € G. An element b E G is called a conjugate of a if there exists an…

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Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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Q: Suppose that G is an abelian group and H 4G. Suppose that every element of H is square, that is,…

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Q: 1. (103) Let G be a group with identity element e, and let H and K be subgroups of G. Assume that…

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Q: Suppose that G is a group and |G| = pnm, where p is prime and p &gt; m. Prove that a Sylow…

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Q: In the group (Z, +), find (-1), the cyclic subgroup generated by -1. Let G be an abelian group, and…

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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: Let G be a group with order n, with n> 2. Prove that G has an element of prime order.

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Q: Suppose H and K are subgroups of a group G. If |H| = 12 and |K| = 35, find |H N K|. Generalize. %3D

A: Given that H and K are subgroups of a group G. Also, the order of H is H=12 and the order of K is…

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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