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1. The problem statement, all variables and given/known data

Let G be a group and g,h in G. Define the commutator of g and h as [g,h]= gh(hg)^-1. Then define the commutator subgroup, denoted [G,G], of G as the subgroup generated by all the commutators of elements of G, i.e. [G,G]=<{[g,h]: g,h in G}>.

(a) What does an element of [G,G] look like?

Hint: it is not enough to only consider elements of the form [g,h].

(b) Prove that [G,G] is a normal subgroup of G.

(c) Prove that G/[G,G] is an Abelian group.

3. The attempt at a solution

(a) Since [G,G] is cyclic, I thought an element would be of some form of [gh(hg)^-1]^n for some n in N. But the hint kind of threw me off...

(b) Initially I wanted to show that gh(hg)^-1 is mm^-1 but I have to show commutativity there and I don't know how to...so there must be another way? I can't seem to get started on this one...

(c) Is this just a consequence of part (b)? Since to show a subgroup is normal you need to use their cosets as an example.

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# Homework Help: Subgroups-commutator, normal, Abelian

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