## Question

###### (#) Lct G b group' . Let G.G denote the swallect subgroup of G containing5 = {ryr 'y :I"â‚¬G}(The subgroup [G,G] called the commutator subgroup of G )Show: thatryr-'w-1, (u-I s")(u-'yu)(u-'ru)-!(u-'yu)-1 for all r,y; eG Deduce that [G,G]<G Show that the quotient group G {{G,G) abelian. I6 NaG and G}N is alxelian, show that (G,G] â‚¬ N. (In other words, G/IG.G] tluee largest alwlian quotient of G.) Slow that il k G a [G.G c k thcn K <6.

(#) Lct G b group' . Let G.G denote the swallect subgroup of G containing 5 = {ryr 'y :I"â‚¬G} (The subgroup [G,G] called the commutator subgroup of G ) Show: that ryr-'w-1, (u-I s")(u-'yu)(u-'ru)-!(u-'yu)-1 for all r,y; eG Deduce that [G,G]<G Show that the quotient group G {{G,G) abelian. I6 NaG and G}N is alxelian, show that (G,G] â‚¬ N. (In other words, G/IG.G] tluee largest alwlian quotient of G.) Slow that il k G a [G.G c k thcn K <6.

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