Question

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)

Answer #1

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

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 G and H be groups, and let
G0 = {(g, 1) : g ∈ G} .
(a) Show that G0 ≅ G.
(b) Show that G0 is a normal subgroup of G × H.
(c) Show that (G × H)/G0 ≅ H.

Let G and G′ be two isomorphic groups that have a unique
normal subgroup of a given
order n, H and H′. Show that the quotient groups G/H and G′/H′
are isomorphic.

Let G be a group and α : G → H be a homomorphism of groups with
H abelian. Show that α factors via G/[G, G], i.e. there exists a
homomorphism β : G/[G, G] −→ H, such that α = β◦q, where q : G −→
G/[G, G] is the quotient homomorphis

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

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

1-Assume that f is a homomorphism from (G, ∗) to (H, ⊗).
a) If a∈G, why is f(a^-1) = (f(a))^−1? (To show that w=z^−1 it
suffices to show that wz=e.)
b) For all a and b in ker(f) why is a∗b∈ker(f)?
c) Assume that z and w are in the range off. Then there are
elements a and b of G such that f(a)=z and f(b)=w. Why is z⊗w in
the range of f?
d) Assume that z is in...

Let H be a subgroup of G, and N be the normalizer of H in G and
C be the centralizer of H in G. Prove that C is normal in N and the
group N/C is isomorphic to a subgroup of Aut(H).

Let N and H be groups, and here for a homomorphism f:
H --> Aut(N) = group automorphism,
let N x_f H be the corresponding semi-direct product.
Let g be in Aut(N), and k be in Aut(H), Let C_g:
Aut(N) --> Aut(N) be given by
conjugation by g.
Now let z := C_g * f * k: H --> Aut(N), where *
means composition.
Show that there is an isomorphism
from Nx_f H to Nx_z H, which takes the natural...

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