Question

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}.

Answer #1

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

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

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 and let H be a subgroup of G Define K
= { g∈ G | g3 ∈ H }. Prove that K is a subgroup of G
.

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

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

Let G be a group and define the center (of G) Z(G) = {a ∈ G | xa
= ax, ∀ x ∈ G}
a. Prove that Z(G) forms a subgroup of G.
b. If G is abelian, show that Z(G) = G.
c. What is the center of S3

: (a) Let p be a prime, and let G be a finite Abelian group.
Show that Gp = {x ∈ G | |x| is a power of p} is a subgroup of G.
(For the identity, remember that 1 = p 0 is a power of p.) (b) Let
p1, . . . , pn be pair-wise distinct primes, and let G be an
Abelian group. Show that Gp1 , . . . , Gpn form direct sum in...

For an abelian group G, let tG = {x E G: x has finite order}
denote its torsion subgroup.
Show that t defines a functor Ab -> Ab if one defines t(f) =
f|tG (f restricted on tG) for every homomorphism f.
If f is injective, then t(f) is injective.
Give an example of a surjective homomorphism f for which t(f)
is not surjective.

Let G be a group and let p be a prime number such that
pg = 0 for every element g ∈ G.
a. If
G is commutative under multiplication, show that the mapping
f : G → G
f(x) =
xp
is a homomorphism
b. If G is
an Abelian group under addition, show that the mapping
f : G → G
f(x) = xpis a homomorphism.

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