Question

Let ? = 〈?, ?|? ^{4} = ? ^{6} = ?, ?? = ??〉. Let
?: ℤ_{4} × ℤ_{6} → ? by ?(c,d) = ? ^{c}?
^{d} for all c ∈ {0,1,2,3} and d∈ {0, 1, 2, 3, 4, 5}.

prove that ? is a homomorphism and if ? is an isomorphism

Answer #1

4. Let f : G→H be a group homomorphism. Suppose a∈G is an
element of finite order n.
(a) Prove that f(a) has finite order k, where k is a divisor of
n.
(b) If f is an isomorphism, prove that k=n.

Let B = { f: ℝ → ℝ
| f is continuous } be the ring of all continuous functions from
the real numbers to the real numbers. Let a be any real number and
define the following function:
Φa:B→R
f(x)↦f(a)
It is called the evaluation homomorphism.
(a) Prove that the evaluation homomorphism is a ring
homomorphism
(b) Describe the image of the evaluation homomorphism.
(c) Describe the kernel of the evaluation homomorphism.
(d) What does the First Isomorphism Theorem for...

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

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

5.
Let S be the set of all polynomials p(x) of degree ≤ 4 such
that
p(-1)=0.
(a) Prove that S is a subspace of the vector space of all
polynomials.
(b) Find a basis for S.
(c) What is the dimension of S?
6.
Let ? ⊆ R! be the span of ?1 = (2,1,0,-1), ?2
=(1,2,-6,1),
?3 = (1,0,2,-1) and ? ⊆ R! be the span of ?1 =(1,1,-2,0), ?2
=(3,1,2,-2). Prove that V=W.

Let
α = 4√ 3 (∈ R), and
consider the homomorphism
ψα : Q[x] → R
f(x) → f(α).
(a) Prove that irr(α, Q) = x^4 −3
(b) Prove that Ker(ψα) = <x^4 −3>
(c) By applying the Fundamental Homomorphism Theorem to ψα,
prove that
L ={a0+a1α+a2α2+a3α3 | a0, a1, a2, a3 ∈ Q }is the smallest
subfield of R containing α.

Let C be a normal subgroup of the group A and let D be a normal
subgroup of the group B.
(a) Prove that C × D is a normal subgroup of A × B
(b) Prove that the map φ : A × B → (A/C) × (B/D) given by φ((m, n))
= (mC, nD) is a group homomorphism.
(c) Use the fundamental homomorphism theorem to prove that (A ×
B)/(C × D) ∼= (A/C) × (B/D)

4. (30) Let C be the ring of complex numbers,and letf:C→C be the
map defined by
f(z) = z^3.
(i) Prove that f is not a homomorphism of rings, by finding an
explicit counterex-
ample.
(ii) Prove that f is not injective.
(iii) Prove that the principal ideal I = 〈x^2 + x + 1〉 is not a
prime ideal of C[x].
(iv) Determine whether or not the ring C[x]/I is a field.

Let G = 〈(1 2 3 4 5 6), (1 6)(2 5)(3 4)〉. Let H1 :=
〈(1 4)(2 5)(3 6)〉 and H2 := 〈(1 6)(2 5)(3 4)〉. Determine
if the subgroups H1 and H2 are normal
subgroups of G.

Let f: Z6 --> Z2 X Z3 be the function given by f([a]6) =
([a]2,[a]3). (a) Show that f is well-defined; that is, show that if
[a]6=[b]6, then f([a]6) = f([b]6). (b) Prove that f is an
isomorphism.

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