Unformatted text preview: ∈ G . ( [ G,G ] is known as the commutator subgroup of G ; it measures how far G is from being abelian. If G is abelian, then xyx1 y1 = e for all x,y ∈ G so [ G,G ] is the trivial group.) (a) Show that [ G,G ] is a normal subgroup of G . (b) Show that G/ [ G,G ] is abelian. (Remark: [ G,G ] is known as the commutator subgroup of G ; it measures how far G is from being abelian. It is the smallest normal subgroup of G such that the quotient is abelian. If G is abelian, then xyx1 y1 = e for all x,y ∈ G so [ G,G ] is the trivial group.) 6. Section 3.2, Exercise 11: Let H ≤ K ≤ G . Prove that  G : H  =  G : K  ·  K : H  (do not assume G is a ﬁnite group). 1...
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 Fall '08
 OGUS
 Algebra, Group Theory, Sets, Normal subgroup, φ

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