Question

Let H be a normal subgroup of G. Assume the quotient group G/H is abelian. Prove that, for any two elements x, y ∈ G, we have x^ (-1) y ^(-1)xy ∈ H

Answer #1

Let G be an Abelian group and H a subgroup of G. Prove that G/H
is Abelian.

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
.

(a) Show that H =<(1234)> is a normal subgroup of G=S4
(b) Is the quotient group G/H abelian? Justify?

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

Let G be a finite group and H be a subgroup of G. Prove that if
H is
only subgroup of G of size |H|, then H is normal in G.

Let H be a subgroup of a group G. Let ∼H and ρH be the
equivalence relation in G introduced in class given by
x∼H y⇐⇒x−1y∈H, xρHy⇐⇒xy−1 ∈H.
The equivalence classes are the left and the right cosets of H in
G, respectively. Prove that the functionφ: G/∼H →G/ρH given
by
φ(xH) = Hx−1
is well-defined and bijective. This proves that the number of
left and right cosets are equal.

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...

a) Let H be a subgroup of a group G satisfying [G ∶ H] = 2. If
there are elements a, b ∈ G such that ab ∈/ H, then prove that
either a ∈ H or b ∈ H. (b) List the left and right cosets of H =
{(1), (23)} in S3. Are they the same collection?

(a) Prove or disprove: Let H and K be two normal subgroups of a
group G. Then the subgroup H ∩ K is normal in G. (b) Prove or
disprove: D4 is normal in S4.

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