Question

Let N1 be a normal subgroup of group G1, and N2 be a normal subgroup of group G2. Use the Fundamental Homomorphism Theorem to show that

G1/N1 × G2/N2 ≃ (G1 × G2)/(N1 × N2).

Answer #1

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)

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 G1 and G2 be isomorphic groups.
Prove that if G1 has a subgroup of order n, then G2 has a
subgroup of order n

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.

Let p be a prime. Show that a group of order
pa has a normal subgroup of order
pb for every nonnegative integer b ≤
a.

1) Let G be a group and N be a normal subgroup. Show that if G
is cyclic, then G/N is cyclic. Is the converse true?
2) What are the zero divisors of Z6?

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

Let G be a finitely generated group, and let H be normal
subgroup of G. Prove that G/H is finitely generated

A subgroup H of a group G is called a normal subgroup if gH=Hg
for all g ∈ G. Every Group contains at least two normal subgroups:
the subgroup consisting of the identity element only {e}; and the
entire group G. If G=S(n) show that A(n) (the subgroup of even
permuations) is also a normal subgroup of G.

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