Question

: (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 G.

Answer #1

Please feel free to ask for any query.

Let G be a non-abelian group of order p^3 with p prime.
(a) Show that |Z(G)| = p. (b) Suppose a /∈ Z(G). Show that
|NG(a)| = p^2 .
(c) Show that G has exactly p 2 +p−1 conjugacy classes (don’t
forget to count the classes of the elements of Z(G)).

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 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 of order p^2, where p is a prime.
Show that G must have a subgroup of order p.
please show with notation if possible

Let G be a finite group and let H, K be normal subgroups of G.
If [G : H] = p and [G : K] = q where p and q are distinct primes,
prove that pq divides [G : H ∩ K].

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.

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 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 finite group and let P be a Sylow p-subgroup of G.
Suppose H is a normal subgroup of G. Prove that HP/H is a Sylow
p-subgroup of G/H and that H ∩ P is a Sylow p-subgroup of H. Hint:
Use the Second Isomorphism theorem.

1. Let p be any prime number. Let r be any integer such that 0
< r < p−1. Show that there exists a number q such that rq =
1(mod p)
2. Let p1 and p2 be two distinct prime numbers. Let r1 and r2 be
such that 0 < r1 < p1 and 0 < r2 < p2. Show that there
exists a number x such that x = r1(mod p1)andx = r2(mod p2).
8. Suppose we roll...

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