Question

Let H <| G. If H is abelian and G/H is also abelian, prove or disprove that G is abelian.

Answer #1

(a) Prove or disprove: if H and K are subgroups of G, then H ∩ K
is a subgroup of G.
(b) Prove or disprove: if H is an abelian subgroup of G, then G
is abelian

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

(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.

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

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 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.

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

Let G be a group of order p^3. Prove that either G is abelian or
its center has exactly p elements.

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.

Suppose that G is a group and H={x|xg=gx for all g∈G}.
a.) Prove that H is a subgroup of G.
b.) Prove that H is abelian.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 23 minutes ago

asked 27 minutes ago

asked 31 minutes ago

asked 49 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago