Question

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.

Answer #1

: (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...

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}.

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.

Let G be a finite group and let H be a subgroup of order n.
Suppose that H is the only subgroup of order n. Show that H is
normal in G.
Hint: Consider the subgroup aHa-1 of G.
Please explain in detail!

let G be a finite group of even order. Show that the equation
x^2=e has even number of solutions in G

Let G be a ﬁnite Abelian group and let n be a positive divisor
of|G|. Show that G has a subgroup of order n.

Let G be a finite group, and suppose that H is normal subgroup
of G.
Show that, for every g ∈ G, the order of gH in G/H must divide
the order of g in G.
What is the order of the coset [4]42 +
〈[6]42〉 in Z42/〈[6]42〉?
Find an example to show that the order of gH in G/H does not
always determine the order of g in G. That is, find an example of a
group G, and...

Let G be an abelian group, let H = {x in G | (x^3) = eg}, where
eg is the identity of G. Prove that H is a subgroup of G.

Let G be a group (not necessarily an Abelian group) of order
425. Prove that G must have an element of order 5. Note, Sylow
Theorem is above us so we can't use it. We're up to Finite Orders.
Thank you.

Let F be a field. It is a general fact that a finite subgroup G
of (F^*,X) of the multiplicative group of a field must be cyclic.
Give a proof by example in the case when |G| = 100.

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