homework5 - G . ( [ G,G ] is known as the commutator...

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Math 113 Section 5 Homework #5 Fall 2010 Due: Monday, October 11 1. Let G be a group and let ( S, · ) be a set with a 2 -to- 1 operation · : S × S S . Let ϕ : G S be a surjective map of sets such that ϕ ( ab ) = ϕ ( a ) · ϕ ( b ) for all a,b G . Show that S is a group under · and that ϕ is a group homomorphism. 2. Let G be a group, and let H be a normal subgroup of G . Let π : G G/H, g Ô→ gH be the natural projection map. Show that ϕ is surjective, satisfies π ( ab ) = π ( a ) · π ( b ) and that π - 1 ( H ) = H . 3. Let G = A 4 and let H = { e, (12)(34) , (13)(24) , (14)(23) } . (a) Show that H is a normal subgroup of G . (b) Show that G/H is isomorphic to Z / 3 Z . 4. Let G = D 2 n be the dihedral group and let H = é r 2 ê . (a) Show that H is a normal subgroup of G . (b) Show that if n is odd, G/H is isomorphic to Z / 2 Z . (c) Show that if n is even, G/H is isomorphic to Z / 2 Z × Z / 2 Z . 5. Let G be a group. Let [ G,G ] be the subgroup generated by elements of the form xyx - 1 y - 1 for all x,y
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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 xyx-1 y-1 = 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 xyx-1 y-1 = 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 nite group). 1...
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