Prove that Z/nZ is a group under the binary operator "+" for every n in positive...

Prove that Z/nZ is a group under the binary operator "+" for every n in positive Z, where Z is the set of integers.

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Colby Messinger answered 2 years ago

Consider the group Z/mZ ⊕ Z/nZ, for any positive integers m and n that are divisible...

Consider the group Z/mZ ⊕ Z/nZ, for any positive integers m and n that are divisible by 4. How many elements of order 4 does G have, and why?

Prove (Z/mZ)/(nZ/mZ) is isomorphic to Z/nZ where n and m are integers greater than 1 and...

Prove (Z/mZ)/(nZ/mZ) is isomorphic to Z/nZ where n and m are integers greater than 1 and n divides m.

Prove or disprove that 3|(n 3 − n) for every positive integer n.

Let the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider...

Let the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider the following statement: for every positive integer n and every x in Z/nZ, there exists y ∈ Z/nZ such that xy = [1]. (a) Write the negation of this statement. (b) Is the original statement true or false? Justify your answer.

Prove that n, nth roots of unity form a group under multiplication.

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...

a) The operator aˆ satisfies the equation [aˆ, aˆ†] = n. Prove that n is a...

a) The operator aˆ satisfies the equation [aˆ, aˆ†] = n. Prove that n is a real number. [Hint: Apply the Hermitian conjugation operation “†” to the above equation.] b) Consider the Hamiltonian Hˆ = h ̄ωˆa†ˆa, where ω > 0 is a real parameter. Prove that Hˆ is Hermitian. c) Let |ψ〉 be an eigenstate of the above Hamiltonian with the energy ε. Use the commutation relations from part a) to prove that aˆ†|ψ〉 is also an eigenstate of...

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 = 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

Let n be a positive integer. Prove that if n is composite, then n has a...

Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )

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