Question

Suppose φ:Q→Z is a homomorphism (both groups are under addition). Prove that φ is the zero map, i.e., φ(x) = 0 for all x ∈ Q.

Answer #1

Suppose G, H be groups and φ : G → H be a group homomorphism.
Then the for any subgroup K of G, the image φ (K) = {y ∈ H | y =
f(x) for some x ∈ G}
is a group a group in H.

Let
φ:G ——H be a group homomorphism and K=ker(φ). Assume that xK=yK.
Prove that φ(x)=φ(y)

Suppose φ : G → G′ is an isomorphism.
(a) Prove that φ(Z(G)) = Z(G′).
(b) Prove that |g| = |φ(g)| for all g ∈ G

Prove the following theorem: Let φ: G→G′ be a group
homomorphism, and let H=ker(φ). Let a∈G.Then the set
(φ)^{-1}[{φ(a)}] ={x∈G|φ(x)} =φ(a)
is the left coset aH of H, and is also the right coset Ha of H.
Consequently, the two partitions of G into left cosets and into
right cosets of H are the same

Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G →
G by φ(x) = x^ k . (a) Prove that φ is a group homomorphism. (b)
Assume that G is finite and |G| is relatively prime to k. Prove
that Ker φ = {e}.

Suppose G and H are groups and ϕ:G -> H is a homomorphism.
Let N be a normal subgroup of G contained in ker(ϕ). Define a
mapping ψ: G/N -> H by ψ (aN)= ϕ (a) for all a in G.
Prove that ψ is a well-defined homomorphism from G/N to H.
Is ψ always an isomorphism? Prove it or give a
counterexample

1. Which of the following sets in (a) are groups under addition?
For each set which is not a group under addition, show which group
property does not apply by counterexample.
a. N; W; Z; Q; R; E; C; P(x, 3); M(2,1,N) .

) In the Extra Problem on PS 2 we defined a map φ:M2,2(Z) →
M2,2(Z2) by the formula φ a b c d = a mod (2) b mod (2) c mod (2) d
mod (2)
On PS 2 you showed that φ is a ring homomorphism and that ker(φ)
= 2a 2b 2c 2d a, b, c, d ∈ Z We know the kernel of any ring
homomorphism is an ideal. Let I = ker(φ).
(a) (6 points) The...

Prove that the set V of all polynomials of degree ≤ n including
the zero polynomial is vector space over the field R under usual
polynomial addition and scalar multiplication. Further, find the
basis for the space of polynomial p(x) of degree ≤ 3. Find a basis
for the subspace with p(1) = 0.

Let R = {0, 2, 4, 6, 8} under addition and multiplication modulo
10. Prove that R is
a field.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 10 minutes ago

asked 15 minutes ago

asked 21 minutes ago

asked 27 minutes ago

asked 32 minutes ago

asked 32 minutes ago

asked 39 minutes ago

asked 39 minutes ago

asked 39 minutes ago

asked 53 minutes ago

asked 1 hour ago

asked 1 hour ago