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)

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove...

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove that xa+b = xaxb b) prove that (xa)-1 = x-a c) establish part a) for arbitrary integers a and b in Z (positive, negative or zero)

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

let n belongs to N and let a, b belong to Z. prove that a is...

let n belongs to N and let a, b belong to Z. prove that a is congruent to b, mod n, if and only if a and b have the same remainder when divided by n.

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 A and B be groups, and consider the product group G=A x B. (a) Prove...

Let A and B be groups, and consider the product group G=A x B. (a) Prove that N={(ea,b) E A x B| b E B} is a subgroup. (b) Prove that N is isomorphic to B (c) Prove that N is a normal subgroup of G (d) Prove that G|N is isomorphic to A

Let G be an abelian group and n a fixed positive integer. Prove that the following...

Let G be an abelian group and n a fixed positive integer. Prove that the following sets are subgroups of G. (a) P(G, n) = {gn | g ∈ G}. (b) T(G, n) = {g ∈ G | gn = 1}. (c) Compute P(G, 2) and T(G, 2) if G = C8 × C2. (d) Prove that T(G, 2) is not a subgroup of G = Dn for n ≥ 3 (i.e the statement above is false when G is...

b belongs to N. Prove that if {7m, m belong to Z} is not belongs to...

b belongs to N. Prove that if {7m, m belong to Z} is not belongs to {ab, a is integer}, then b =1

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.

Let f : Z × Z → Z be defined by f(n, m) = n −...

Let f : Z × Z → Z be defined by f(n, m) = n − m a. Is this function one to one? Prove your result. b. Is this function onto Z? Prove your result

Let A = Z and let a, b ∈ A. Prove if the following binary operations...

Let A = Z and let a, b ∈ A. Prove if the following binary operations are (i) commutative, (2) if they are associative and (3) if they have an identity (if the operations has an identity, give the identity or show that the operation has no identity). (a) (3 points) f(a, b) = a + b + 1 (b) (3 points) f(a, b) = a(b + 1) (c) (3 points) f(a, b) = x2 + xy + y2

Subjects