Let G be a group of order mn where gcd(m,n)=1 Let a and b be elements...

Let G be a group of order mn where gcd(m,n)=1

Let a and b be elements in G such that o(a)=m and 0(b)=n

Prove that G is cyclic if and only if ab=ba

Expert Solution

Colby Messinger answered 2 years ago

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove...

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove (a). (a^m)(a^n)=a^(m+n) (b). (a^m)^n=a^(mn)

Let A be a set with m elements and B a set of n elements, where...

Let A be a set with m elements and B a set of n elements, where m; n are positive integers. Find the number of one-to-one functions from A to B.

Let m, n be natural numbers such that their greatest common divisor gcd(m, n) = 1....

Let m, n be natural numbers such that their greatest common divisor gcd(m, n) = 1. Prove that there is a natural number k such that n divides ((m^k) − 1).

Let A[1 · · · n] be an array of n elements and B[1 · ·...

Let A[1 · · · n] be an array of n elements and B[1 · · · m] an array of m elements. We assume that m ≤ n. Note that neither A nor B is sorted. The problem is to compute the number of elements of A that are smaller than B[i] for each element B[i] with 1 ≤ i ≤ m. For example, let A be {30, 20, 100, 60, 90, 10, 40, 50, 80, 70} of ten...

Let A[1 · · · n] be an array of n elements and B[1 · ·...

Let A[1 · · · n] be an array of n elements and B[1 · · · m] an array of m elements. We assume that m ≤ n. Note that neither A nor B is sorted. The problem is to compute the number of elements of A that are smaller than B[i] for each element B[i] with 1 ≤ i ≤ m. For example, let A be {30, 20, 100, 60, 90, 10, 40, 50, 80, 70} of ten...

Let G be a group of order p am where p is a prime not dividing...

Let G be a group of order p am where p is a prime not dividing m. Show the following 1. Sylow p-subgroups of G exist; i.e. Sylp(G) 6= ∅. 2. If P ∈ Sylp(G) and Q is any p-subgroup of G, then there exists g ∈ G such that Q 6 gP g−1 ; i.e. Q is contained in some conjugate of P. In particular, any two Sylow p- subgroups of G are conjugate in G. 3. np ≡...

Let A∈Mn(R)"> A ∈ M n ( R ) A∈Mn(R) such that I+A"> I + A I+A is invertible. Suppose that

Let A∈Mn(R)A∈Mn(R) such that I+AI+A is invertible. Suppose thatB=(I−A)(I+A)−1B=(I−A)(I+A)−1(a) Show that B=(I+A)−1(I−A)B=(I+A)−1(I−A) (b) Show that I+BI+B is invertible and express AA in terms of BB.

abstract algebra Let G be a finite abelian group of order n Prove that if d...

abstract algebra Let G be a finite abelian group of order n Prove that if d is a positive divisor of n, then G has a subgroup of order d.

1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) = At. Compute the...

1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) = At. Compute the characteristic polynomial of T. You may wish to use the basis of Mn×n(F) consisting of the matrices eij + eji, eij −eji and eii. 2. Let A = (a b c d) (2 by 2 matrix) and let T : M2×2(F) → M2×2(F) be deﬁned asT (B) = AB. Represent T as a 4×4 matrix using the ordered basis {e11,e21,e12,e22}, and use this matrix to...

Let a be an element of a finite group G. The order of a is the...

Let a be an element of a finite group G. The order of a is the least power k such that ak = e. Find the orders of following elements in S5 a. (1 2 3 ) b. (1 3 2 4) c. (2 3) (1 4) d. (1 2) (3 5 4)

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Let G be a group of order mn where gcd(m,n)=1 Let a and b be elements...

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