Question

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

Answer #1

Let φ : A → B be a group homomorphism. Prove that ker φ is a
normal subgroup of A.

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}.

Letφ:G→G′is a group homomorphism. Prove that φ is one-to-one if
and only if Ker(φ) ={e}.

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.

Please explain it in detail.
Let φ∶G → H be a homomorphism with H abelian. Show that G/ ker φ
must be abelian.

Let G and H be groups and f:G--->H be a surjective
homomorphism. Let J be a subgroup of H and define f^-1(J) ={x is an
element of G| f(x) is an element of J}
a. Show ker(f)⊂f^-1(J) and ker(f) is a normal subgroup of
f^-1(J)
b. Let p: f^-1(J) --> J be defined by p(x) = f(x). Show p is
a surjective homomorphism
c. Show the set kef(f) and ker(p) are equal
d. Show J is isomorphic to f^-1(J)/ker(f)

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

Let φ : G → G′ be an onto homomorphism and let N be a normal
subgroup of G. Prove that φ(N) is a normal subgroup of G′.

Prove this statement: Let ϕ : A1 → A2 be a homomorphism and let
N = ker ϕ. Then A1/N is isomorphic to ϕ(A1). Further ψ : A1/N →
ϕ(A1) defined by ψ(aN) = ϕ(a) is an isomorphism.
You must use the following elements to prove:
- well-definedness
- one-to-one
- onto
- homomorphism

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