Question

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

Answer #1

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

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 group, and H a subgroup of G, let a,b?G prove the
statement or give a counterexample:
If aH=bH, then Ha=Hb

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.

let g be a group. let h be a subgroup of g. define
a~b. if ab^-1 is in h. prove ~ is an equivalence relation on g

Abstract Algebra (Modern Algebra)
Prove that every subgroup of an abelian group is abelian.

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